lecture 3.2, math-57091 probability and statistics for ...zvavitch/57091_lecture_3_2_2014.pdflecture...
TRANSCRIPT
![Page 1: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/1.jpg)
Lecture 3.2,MATH-57091 Probability and Statistics for High-School
Teachers.
Artem Zvavitch
Department of Mathematical Sciences, Kent State University
September, 8 - 12, 2014.
![Page 2: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/2.jpg)
Discrete Random Variables
A random variable that can take at most a countable (finite) set of possible values issaid to be discrete.
In this case we may define a probability mass function, i.e. theprobability that a discrete random variable is exactly equal to some value. Moreprecisely X takes value x1,x2, . . . such that
p(xi ) = P(X = xi ),
where
p(xi ) ∈ [0,1] and∞∑i=1
p(xi ) = 1.
If X is a discrete Random variable that take one of the possible values x1,x2, . . . ,xnthen the expected value of X , denoted by EX is defined by
EX =
n∑i=1
xi P(X = xi ) =∑
xi p(xi ).
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 3: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/3.jpg)
Discrete Random Variables
A random variable that can take at most a countable (finite) set of possible values issaid to be discrete. In this case we may define a probability mass function, i.e. theprobability that a discrete random variable is exactly equal to some value.
Moreprecisely X takes value x1,x2, . . . such that
p(xi ) = P(X = xi ),
where
p(xi ) ∈ [0,1] and∞∑i=1
p(xi ) = 1.
If X is a discrete Random variable that take one of the possible values x1,x2, . . . ,xnthen the expected value of X , denoted by EX is defined by
EX =
n∑i=1
xi P(X = xi ) =∑
xi p(xi ).
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 4: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/4.jpg)
Discrete Random Variables
A random variable that can take at most a countable (finite) set of possible values issaid to be discrete. In this case we may define a probability mass function, i.e. theprobability that a discrete random variable is exactly equal to some value. Moreprecisely X takes value x1,x2, . . . such that
p(xi ) = P(X = xi ),
where
p(xi ) ∈ [0,1] and∞∑i=1
p(xi ) = 1.
If X is a discrete Random variable that take one of the possible values x1,x2, . . . ,xnthen the expected value of X , denoted by EX is defined by
EX =
n∑i=1
xi P(X = xi ) =∑
xi p(xi ).
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 5: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/5.jpg)
Discrete Random Variables
A random variable that can take at most a countable (finite) set of possible values issaid to be discrete. In this case we may define a probability mass function, i.e. theprobability that a discrete random variable is exactly equal to some value. Moreprecisely X takes value x1,x2, . . . such that
p(xi ) = P(X = xi ),
where
p(xi ) ∈ [0,1] and∞∑i=1
p(xi ) = 1.
If X is a discrete Random variable that take one of the possible values x1,x2, . . . ,xnthen the expected value of X , denoted by EX is defined by
EX =
n∑i=1
xi P(X = xi ) =∑
xi p(xi ).
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 6: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/6.jpg)
Bernoulli Random Variable
Suppose that a trial or an experiment, whose outcome can be classified as either a"success" or as "failure", is performed.
If we let X equal 1 if the outcome us asuccess and 0 if it is a failure then the probability mass function is given by
p(0) = P(X = 0) = 1−p,
p(1) = P(X = 1) = p,
where p ∈ [0,1] is a probability that the trial is a "success". A random variable X issaid to be a Bernoulli random variable if its as above and p ∈ (0,1).
Expectation of Bernoulli Random VariableIt is not so hard to compute EX if X is a Bernoulli Random variable with parameter p:
EX = 0∗ (1−p)+1∗p = p.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 7: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/7.jpg)
Bernoulli Random Variable
Suppose that a trial or an experiment, whose outcome can be classified as either a"success" or as "failure", is performed. If we let X equal 1 if the outcome us asuccess and 0 if it is a failure then the probability mass function is given by
p(0) = P(X = 0) = 1−p,
p(1) = P(X = 1) = p,
where p ∈ [0,1] is a probability that the trial is a "success".
A random variable X issaid to be a Bernoulli random variable if its as above and p ∈ (0,1).
Expectation of Bernoulli Random VariableIt is not so hard to compute EX if X is a Bernoulli Random variable with parameter p:
EX = 0∗ (1−p)+1∗p = p.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 8: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/8.jpg)
Bernoulli Random Variable
Suppose that a trial or an experiment, whose outcome can be classified as either a"success" or as "failure", is performed. If we let X equal 1 if the outcome us asuccess and 0 if it is a failure then the probability mass function is given by
p(0) = P(X = 0) = 1−p,
p(1) = P(X = 1) = p,
where p ∈ [0,1] is a probability that the trial is a "success". A random variable X issaid to be a Bernoulli random variable if its as above and p ∈ (0,1).
Expectation of Bernoulli Random VariableIt is not so hard to compute EX if X is a Bernoulli Random variable with parameter p:
EX = 0∗ (1−p)+1∗p = p.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 9: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/9.jpg)
Bernoulli Random Variable
Suppose that a trial or an experiment, whose outcome can be classified as either a"success" or as "failure", is performed. If we let X equal 1 if the outcome us asuccess and 0 if it is a failure then the probability mass function is given by
p(0) = P(X = 0) = 1−p,
p(1) = P(X = 1) = p,
where p ∈ [0,1] is a probability that the trial is a "success". A random variable X issaid to be a Bernoulli random variable if its as above and p ∈ (0,1).
Expectation of Bernoulli Random VariableIt is not so hard to compute EX if X is a Bernoulli Random variable with parameter p:
EX = 0∗ (1−p)+1∗p = p.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 10: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/10.jpg)
The Binomial Random VariableSuppose that n independent trial, each of which results in a "success" with probabilityp and "failure" with probability 1−p are to be performed.
If X represents thenumber of successes that occur in the n trials, then X is said to be a BinomialRandom Variable with parameters (n,p). We note that
p(i) = P(X = i) =(n
i)
pi (1−p)n−i , for i = 0,1, . . . ,n.
where (ni)=
n!(n− i)!i!
.
Please, note that the above formula is not so hard to verify! First notice that forparticular sequence of n outcomes to have exactly i successes and thus n− i failures isexactly pi (1−p)n−i . But we have a lot of ways to select "when" our i successes occurand the number of those "ways" is exactly
(ni
). It is also a good moment to remind
the Binomial Formula
(x + y)n =
n∑i=0
(ni)
x i yn−i .
One can see the formula just from noticing that (x + y)n is (x + y) multiplied ntimes. Thus when we open the parentheses you need to select x or y from eachmultiplier only once so to get x i yn−i you need to pick i - x and (n− i) - y and youcan do it in
(ni
)ways. The binomial formula immediately gives that
n∑i=0
P(X = i) =n∑
i=0
(ni)
pi (1−p)n−i = (p+(1−p))n = 1.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 11: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/11.jpg)
The Binomial Random VariableSuppose that n independent trial, each of which results in a "success" with probabilityp and "failure" with probability 1−p are to be performed. If X represents thenumber of successes that occur in the n trials, then X is said to be a BinomialRandom Variable with parameters (n,p).
We note that
p(i) = P(X = i) =(n
i)
pi (1−p)n−i , for i = 0,1, . . . ,n.
where (ni)=
n!(n− i)!i!
.
Please, note that the above formula is not so hard to verify! First notice that forparticular sequence of n outcomes to have exactly i successes and thus n− i failures isexactly pi (1−p)n−i . But we have a lot of ways to select "when" our i successes occurand the number of those "ways" is exactly
(ni
). It is also a good moment to remind
the Binomial Formula
(x + y)n =
n∑i=0
(ni)
x i yn−i .
One can see the formula just from noticing that (x + y)n is (x + y) multiplied ntimes. Thus when we open the parentheses you need to select x or y from eachmultiplier only once so to get x i yn−i you need to pick i - x and (n− i) - y and youcan do it in
(ni
)ways. The binomial formula immediately gives that
n∑i=0
P(X = i) =n∑
i=0
(ni)
pi (1−p)n−i = (p+(1−p))n = 1.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 12: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/12.jpg)
The Binomial Random VariableSuppose that n independent trial, each of which results in a "success" with probabilityp and "failure" with probability 1−p are to be performed. If X represents thenumber of successes that occur in the n trials, then X is said to be a BinomialRandom Variable with parameters (n,p). We note that
p(i) = P(X = i) =(n
i)
pi (1−p)n−i , for i = 0,1, . . . ,n.
where (ni)=
n!(n− i)!i!
.
Please, note that the above formula is not so hard to verify! First notice that forparticular sequence of n outcomes to have exactly i successes and thus n− i failures isexactly pi (1−p)n−i . But we have a lot of ways to select "when" our i successes occurand the number of those "ways" is exactly
(ni
). It is also a good moment to remind
the Binomial Formula
(x + y)n =
n∑i=0
(ni)
x i yn−i .
One can see the formula just from noticing that (x + y)n is (x + y) multiplied ntimes. Thus when we open the parentheses you need to select x or y from eachmultiplier only once so to get x i yn−i you need to pick i - x and (n− i) - y and youcan do it in
(ni
)ways. The binomial formula immediately gives that
n∑i=0
P(X = i) =n∑
i=0
(ni)
pi (1−p)n−i = (p+(1−p))n = 1.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 13: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/13.jpg)
The Binomial Random VariableSuppose that n independent trial, each of which results in a "success" with probabilityp and "failure" with probability 1−p are to be performed. If X represents thenumber of successes that occur in the n trials, then X is said to be a BinomialRandom Variable with parameters (n,p). We note that
p(i) = P(X = i) =(n
i)
pi (1−p)n−i , for i = 0,1, . . . ,n.
where (ni)=
n!(n− i)!i!
.
Please, note that the above formula is not so hard to verify! First notice that forparticular sequence of n outcomes to have exactly i successes and thus n− i failures isexactly pi (1−p)n−i . But we have a lot of ways to select "when" our i successes occurand the number of those "ways" is exactly
(ni
).
It is also a good moment to remindthe Binomial Formula
(x + y)n =
n∑i=0
(ni)
x i yn−i .
One can see the formula just from noticing that (x + y)n is (x + y) multiplied ntimes. Thus when we open the parentheses you need to select x or y from eachmultiplier only once so to get x i yn−i you need to pick i - x and (n− i) - y and youcan do it in
(ni
)ways. The binomial formula immediately gives that
n∑i=0
P(X = i) =n∑
i=0
(ni)
pi (1−p)n−i = (p+(1−p))n = 1.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 14: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/14.jpg)
The Binomial Random VariableSuppose that n independent trial, each of which results in a "success" with probabilityp and "failure" with probability 1−p are to be performed. If X represents thenumber of successes that occur in the n trials, then X is said to be a BinomialRandom Variable with parameters (n,p). We note that
p(i) = P(X = i) =(n
i)
pi (1−p)n−i , for i = 0,1, . . . ,n.
where (ni)=
n!(n− i)!i!
.
Please, note that the above formula is not so hard to verify! First notice that forparticular sequence of n outcomes to have exactly i successes and thus n− i failures isexactly pi (1−p)n−i . But we have a lot of ways to select "when" our i successes occurand the number of those "ways" is exactly
(ni
). It is also a good moment to remind
the Binomial Formula
(x + y)n =
n∑i=0
(ni)
x i yn−i .
One can see the formula just from noticing that (x + y)n is (x + y) multiplied ntimes. Thus when we open the parentheses you need to select x or y from eachmultiplier only once so to get x i yn−i you need to pick i - x and (n− i) - y and youcan do it in
(ni
)ways. The binomial formula immediately gives that
n∑i=0
P(X = i) =n∑
i=0
(ni)
pi (1−p)n−i = (p+(1−p))n = 1.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 15: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/15.jpg)
The Binomial Random VariableSuppose that n independent trial, each of which results in a "success" with probabilityp and "failure" with probability 1−p are to be performed. If X represents thenumber of successes that occur in the n trials, then X is said to be a BinomialRandom Variable with parameters (n,p). We note that
p(i) = P(X = i) =(n
i)
pi (1−p)n−i , for i = 0,1, . . . ,n.
where (ni)=
n!(n− i)!i!
.
Please, note that the above formula is not so hard to verify! First notice that forparticular sequence of n outcomes to have exactly i successes and thus n− i failures isexactly pi (1−p)n−i . But we have a lot of ways to select "when" our i successes occurand the number of those "ways" is exactly
(ni
). It is also a good moment to remind
the Binomial Formula
(x + y)n =
n∑i=0
(ni)
x i yn−i .
One can see the formula just from noticing that (x + y)n is (x + y) multiplied ntimes. Thus when we open the parentheses you need to select x or y from eachmultiplier only once so to get x i yn−i you need to pick i - x and (n− i) - y and youcan do it in
(ni
)ways.
The binomial formula immediately gives thatn∑
i=0
P(X = i) =n∑
i=0
(ni)
pi (1−p)n−i = (p+(1−p))n = 1.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 16: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/16.jpg)
The Binomial Random VariableSuppose that n independent trial, each of which results in a "success" with probabilityp and "failure" with probability 1−p are to be performed. If X represents thenumber of successes that occur in the n trials, then X is said to be a BinomialRandom Variable with parameters (n,p). We note that
p(i) = P(X = i) =(n
i)
pi (1−p)n−i , for i = 0,1, . . . ,n.
where (ni)=
n!(n− i)!i!
.
Please, note that the above formula is not so hard to verify! First notice that forparticular sequence of n outcomes to have exactly i successes and thus n− i failures isexactly pi (1−p)n−i . But we have a lot of ways to select "when" our i successes occurand the number of those "ways" is exactly
(ni
). It is also a good moment to remind
the Binomial Formula
(x + y)n =
n∑i=0
(ni)
x i yn−i .
One can see the formula just from noticing that (x + y)n is (x + y) multiplied ntimes. Thus when we open the parentheses you need to select x or y from eachmultiplier only once so to get x i yn−i you need to pick i - x and (n− i) - y and youcan do it in
(ni
)ways. The binomial formula immediately gives that
n∑i=0
P(X = i) =n∑
i=0
(ni)
pi (1−p)n−i = (p+(1−p))n = 1.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 17: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/17.jpg)
The Binomial Random Variable: Examples
Questions: Four fair coins are flipped. If the outcomes are assumes independent, whatis the probability that two heads and two tails are obtained?
Solution: Letting Xequal the number of heads ("successes") that appear, then X is a binomial randomvariable with parameters n = 4,p = 1
2 . So
P(X = 2) =(42)(1
2
)2(12
)2=
38
.
Questions: It is known that all computers produced by BRC will be defective withprobability .2, independently of each other. What is the probability that in a sample offour computers, at most 1 is defective?Solution: Let X be a number of defective items in the sample, then X is a binomialrandom variable with parameters (4, .2). Thus, the desired probability is given by
P(X = 0)+P(X = 1) =(40)
.20.84+(41)
.21.83 = .8192
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 18: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/18.jpg)
The Binomial Random Variable: Examples
Questions: Four fair coins are flipped. If the outcomes are assumes independent, whatis the probability that two heads and two tails are obtained? Solution: Letting Xequal the number of heads ("successes") that appear, then X is a binomial randomvariable with parameters n = 4,p = 1
2 .
So
P(X = 2) =(42)(1
2
)2(12
)2=
38
.
Questions: It is known that all computers produced by BRC will be defective withprobability .2, independently of each other. What is the probability that in a sample offour computers, at most 1 is defective?Solution: Let X be a number of defective items in the sample, then X is a binomialrandom variable with parameters (4, .2). Thus, the desired probability is given by
P(X = 0)+P(X = 1) =(40)
.20.84+(41)
.21.83 = .8192
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 19: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/19.jpg)
The Binomial Random Variable: Examples
Questions: Four fair coins are flipped. If the outcomes are assumes independent, whatis the probability that two heads and two tails are obtained? Solution: Letting Xequal the number of heads ("successes") that appear, then X is a binomial randomvariable with parameters n = 4,p = 1
2 . So
P(X = 2) =(42)(1
2
)2(12
)2=
38
.
Questions: It is known that all computers produced by BRC will be defective withprobability .2, independently of each other. What is the probability that in a sample offour computers, at most 1 is defective?Solution: Let X be a number of defective items in the sample, then X is a binomialrandom variable with parameters (4, .2). Thus, the desired probability is given by
P(X = 0)+P(X = 1) =(40)
.20.84+(41)
.21.83 = .8192
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 20: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/20.jpg)
The Binomial Random Variable: Examples
Questions: Four fair coins are flipped. If the outcomes are assumes independent, whatis the probability that two heads and two tails are obtained? Solution: Letting Xequal the number of heads ("successes") that appear, then X is a binomial randomvariable with parameters n = 4,p = 1
2 . So
P(X = 2) =(42)(1
2
)2(12
)2=
38
.
Questions: It is known that all computers produced by BRC will be defective withprobability .2, independently of each other. What is the probability that in a sample offour computers, at most 1 is defective?
Solution: Let X be a number of defective items in the sample, then X is a binomialrandom variable with parameters (4, .2). Thus, the desired probability is given by
P(X = 0)+P(X = 1) =(40)
.20.84+(41)
.21.83 = .8192
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 21: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/21.jpg)
The Binomial Random Variable: Examples
Questions: Four fair coins are flipped. If the outcomes are assumes independent, whatis the probability that two heads and two tails are obtained? Solution: Letting Xequal the number of heads ("successes") that appear, then X is a binomial randomvariable with parameters n = 4,p = 1
2 . So
P(X = 2) =(42)(1
2
)2(12
)2=
38
.
Questions: It is known that all computers produced by BRC will be defective withprobability .2, independently of each other. What is the probability that in a sample offour computers, at most 1 is defective?Solution: Let X be a number of defective items in the sample, then X is a binomialrandom variable with parameters (4, .2).
Thus, the desired probability is given by
P(X = 0)+P(X = 1) =(40)
.20.84+(41)
.21.83 = .8192
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 22: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/22.jpg)
The Binomial Random Variable: Examples
Questions: Four fair coins are flipped. If the outcomes are assumes independent, whatis the probability that two heads and two tails are obtained? Solution: Letting Xequal the number of heads ("successes") that appear, then X is a binomial randomvariable with parameters n = 4,p = 1
2 . So
P(X = 2) =(42)(1
2
)2(12
)2=
38
.
Questions: It is known that all computers produced by BRC will be defective withprobability .2, independently of each other. What is the probability that in a sample offour computers, at most 1 is defective?Solution: Let X be a number of defective items in the sample, then X is a binomialrandom variable with parameters (4, .2). Thus, the desired probability is given by
P(X = 0)+P(X = 1) =(40)
.20.84+(41)
.21.83 = .8192
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 23: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/23.jpg)
The Binomial Random Variable: Expectation
Let X be a binomial random variable with parameters (n,p) then
EX =
n∑i=0
i(n
i
)pi (1− p)n−i =
n∑i=1
in!
(n − i)!i!pi (1− p)n−i
=
n∑i=1
n!(n − i)!(i − 1)!
pi (1− p)n−i
= npn∑
i=1
(n − 1)!((n − 1)− (i − 1))!(i − 1)!
pi−1(1− p)(n−1)−(i−1)
= npn−1∑j=0
(n − 1)!((n − 1)− j)!j!
pj (1− p)(n−1)−j
= npn−1∑j=0
((n − 1)j
)pj (1− p)(n−1)−j
= np(p +(1− p))(n−1)
= np
The above is fun combinatorial way to compute expectation of Binomial Random Variable - soonwe will learn a probabilistic way to do it, it will be much more simpler!
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 24: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/24.jpg)
The Binomial Random Variable: Expectation
Let X be a binomial random variable with parameters (n,p) then
EX =
n∑i=0
i(n
i
)pi (1− p)n−i =
n∑i=1
in!
(n − i)!i!pi (1− p)n−i
=
n∑i=1
n!(n − i)!(i − 1)!
pi (1− p)n−i
= npn∑
i=1
(n − 1)!((n − 1)− (i − 1))!(i − 1)!
pi−1(1− p)(n−1)−(i−1)
= npn−1∑j=0
(n − 1)!((n − 1)− j)!j!
pj (1− p)(n−1)−j
= npn−1∑j=0
((n − 1)j
)pj (1− p)(n−1)−j
= np(p +(1− p))(n−1)
= np
The above is fun combinatorial way to compute expectation of Binomial Random Variable - soonwe will learn a probabilistic way to do it, it will be much more simpler!
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 25: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/25.jpg)
The Binomial Random Variable: Expectation
Let X be a binomial random variable with parameters (n,p) then
EX =
n∑i=0
i(n
i
)pi (1− p)n−i =
n∑i=1
in!
(n − i)!i!pi (1− p)n−i
=
n∑i=1
n!(n − i)!(i − 1)!
pi (1− p)n−i
= npn∑
i=1
(n − 1)!((n − 1)− (i − 1))!(i − 1)!
pi−1(1− p)(n−1)−(i−1)
= npn−1∑j=0
(n − 1)!((n − 1)− j)!j!
pj (1− p)(n−1)−j
= npn−1∑j=0
((n − 1)j
)pj (1− p)(n−1)−j
= np(p +(1− p))(n−1)
= np
The above is fun combinatorial way to compute expectation of Binomial Random Variable - soonwe will learn a probabilistic way to do it, it will be much more simpler!
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 26: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/26.jpg)
The Binomial Random Variable: Expectation
Let X be a binomial random variable with parameters (n,p) then
EX =
n∑i=0
i(n
i
)pi (1− p)n−i =
n∑i=1
in!
(n − i)!i!pi (1− p)n−i
=
n∑i=1
n!(n − i)!(i − 1)!
pi (1− p)n−i
= npn∑
i=1
(n − 1)!((n − 1)− (i − 1))!(i − 1)!
pi−1(1− p)(n−1)−(i−1)
= npn−1∑j=0
(n − 1)!((n − 1)− j)!j!
pj (1− p)(n−1)−j
= npn−1∑j=0
((n − 1)j
)pj (1− p)(n−1)−j
= np(p +(1− p))(n−1)
= np
The above is fun combinatorial way to compute expectation of Binomial Random Variable - soonwe will learn a probabilistic way to do it, it will be much more simpler!
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 27: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/27.jpg)
The Binomial Random Variable: Expectation
Let X be a binomial random variable with parameters (n,p) then
EX =
n∑i=0
i(n
i
)pi (1− p)n−i =
n∑i=1
in!
(n − i)!i!pi (1− p)n−i
=
n∑i=1
n!(n − i)!(i − 1)!
pi (1− p)n−i
= npn∑
i=1
(n − 1)!((n − 1)− (i − 1))!(i − 1)!
pi−1(1− p)(n−1)−(i−1)
= npn−1∑j=0
(n − 1)!((n − 1)− j)!j!
pj (1− p)(n−1)−j
= npn−1∑j=0
((n − 1)j
)pj (1− p)(n−1)−j
= np(p +(1− p))(n−1)
= np
The above is fun combinatorial way to compute expectation of Binomial Random Variable - soonwe will learn a probabilistic way to do it, it will be much more simpler!
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 28: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/28.jpg)
The Binomial Random Variable: Expectation
Let X be a binomial random variable with parameters (n,p) then
EX =
n∑i=0
i(n
i
)pi (1− p)n−i =
n∑i=1
in!
(n − i)!i!pi (1− p)n−i
=
n∑i=1
n!(n − i)!(i − 1)!
pi (1− p)n−i
= npn∑
i=1
(n − 1)!((n − 1)− (i − 1))!(i − 1)!
pi−1(1− p)(n−1)−(i−1)
= npn−1∑j=0
(n − 1)!((n − 1)− j)!j!
pj (1− p)(n−1)−j
= npn−1∑j=0
((n − 1)j
)pj (1− p)(n−1)−j
= np(p +(1− p))(n−1)
= np
The above is fun combinatorial way to compute expectation of Binomial Random Variable - soonwe will learn a probabilistic way to do it, it will be much more simpler!
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 29: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/29.jpg)
The Binomial Random Variable: Expectation
Let X be a binomial random variable with parameters (n,p) then
EX =
n∑i=0
i(n
i
)pi (1− p)n−i =
n∑i=1
in!
(n − i)!i!pi (1− p)n−i
=
n∑i=1
n!(n − i)!(i − 1)!
pi (1− p)n−i
= npn∑
i=1
(n − 1)!((n − 1)− (i − 1))!(i − 1)!
pi−1(1− p)(n−1)−(i−1)
= npn−1∑j=0
(n − 1)!((n − 1)− j)!j!
pj (1− p)(n−1)−j
= npn−1∑j=0
((n − 1)j
)pj (1− p)(n−1)−j
= np(p +(1− p))(n−1)
= np
The above is fun combinatorial way to compute expectation of Binomial Random Variable - soonwe will learn a probabilistic way to do it, it will be much more simpler!
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 30: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/30.jpg)
The Binomial Random Variable: Expectation
Let X be a binomial random variable with parameters (n,p) then
EX =
n∑i=0
i(n
i
)pi (1− p)n−i =
n∑i=1
in!
(n − i)!i!pi (1− p)n−i
=
n∑i=1
n!(n − i)!(i − 1)!
pi (1− p)n−i
= npn∑
i=1
(n − 1)!((n − 1)− (i − 1))!(i − 1)!
pi−1(1− p)(n−1)−(i−1)
= npn−1∑j=0
(n − 1)!((n − 1)− j)!j!
pj (1− p)(n−1)−j
= npn−1∑j=0
((n − 1)j
)pj (1− p)(n−1)−j
= np(p +(1− p))(n−1)
= np
The above is fun combinatorial way to compute expectation of Binomial Random Variable - soonwe will learn a probabilistic way to do it, it will be much more simpler!
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 31: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/31.jpg)
The Binomial Random Variable: Expectation
Let X be a binomial random variable with parameters (n,p) then
EX =
n∑i=0
i(n
i
)pi (1− p)n−i =
n∑i=1
in!
(n − i)!i!pi (1− p)n−i
=
n∑i=1
n!(n − i)!(i − 1)!
pi (1− p)n−i
= npn∑
i=1
(n − 1)!((n − 1)− (i − 1))!(i − 1)!
pi−1(1− p)(n−1)−(i−1)
= npn−1∑j=0
(n − 1)!((n − 1)− j)!j!
pj (1− p)(n−1)−j
= npn−1∑j=0
((n − 1)j
)pj (1− p)(n−1)−j
= np(p +(1− p))(n−1)
= np
The above is fun combinatorial way to compute expectation of Binomial Random Variable - soonwe will learn a probabilistic way to do it, it will be much more simpler!
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 32: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/32.jpg)
The Geometric Random Variable
Suppose that independent trial, each of which results in a "success" with probability pand "failure" with probability 1−p are to be performed until success occurs.
If Xrepresents the number of trial required until the first success, then X is said to be aGeometric Random Variable with parameters p. We note that
p(n) = P(X = n) = (1−p)n−1p, for n = 1,2,3, . . . ,
(please, explain why!).
Please, note that:∞∑
n=1
p(n) =∞∑
n=1
(1−p)n−1p = p∞∑
n=1
(1−p)n−1
= p∞∑
n=0
(1−p)n = p1
1− (1−p)= 1
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 33: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/33.jpg)
The Geometric Random Variable
Suppose that independent trial, each of which results in a "success" with probability pand "failure" with probability 1−p are to be performed until success occurs. If Xrepresents the number of trial required until the first success, then X is said to be aGeometric Random Variable with parameters p.
We note that
p(n) = P(X = n) = (1−p)n−1p, for n = 1,2,3, . . . ,
(please, explain why!).
Please, note that:∞∑
n=1
p(n) =∞∑
n=1
(1−p)n−1p = p∞∑
n=1
(1−p)n−1
= p∞∑
n=0
(1−p)n = p1
1− (1−p)= 1
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 34: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/34.jpg)
The Geometric Random Variable
Suppose that independent trial, each of which results in a "success" with probability pand "failure" with probability 1−p are to be performed until success occurs. If Xrepresents the number of trial required until the first success, then X is said to be aGeometric Random Variable with parameters p. We note that
p(n) = P(X = n) = (1−p)n−1p, for n = 1,2,3, . . . ,
(please, explain why!).
Please, note that:∞∑
n=1
p(n) =∞∑
n=1
(1−p)n−1p = p∞∑
n=1
(1−p)n−1
= p∞∑
n=0
(1−p)n = p1
1− (1−p)= 1
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 35: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/35.jpg)
The Geometric Random Variable
Suppose that independent trial, each of which results in a "success" with probability pand "failure" with probability 1−p are to be performed until success occurs. If Xrepresents the number of trial required until the first success, then X is said to be aGeometric Random Variable with parameters p. We note that
p(n) = P(X = n) = (1−p)n−1p, for n = 1,2,3, . . . ,
(please, explain why!).
Please, note that:∞∑
n=1
p(n) =∞∑
n=1
(1−p)n−1p = p∞∑
n=1
(1−p)n−1
= p∞∑
n=0
(1−p)n = p1
1− (1−p)= 1
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 36: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/36.jpg)
The Geometric Random Variable
Suppose that independent trial, each of which results in a "success" with probability pand "failure" with probability 1−p are to be performed until success occurs. If Xrepresents the number of trial required until the first success, then X is said to be aGeometric Random Variable with parameters p. We note that
p(n) = P(X = n) = (1−p)n−1p, for n = 1,2,3, . . . ,
(please, explain why!).
Please, note that:∞∑
n=1
p(n) =∞∑
n=1
(1−p)n−1p = p∞∑
n=1
(1−p)n−1
= p∞∑
n=0
(1−p)n = p1
1− (1−p)= 1
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 37: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/37.jpg)
Expectation of Geometric Random VariableTo compute EX for geometric random variable X one need to use quite a nice analytictrick.
First we again remind the formula of the sum of geometric progression
∞∑n=0
xn =1
1− x, for x ∈ (0,1).
Then taking the derivatives from both sides we get
ddx
∞∑n=0
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=0
ddx
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=1
nxn−1 =1
(1− x)2, for x ∈ (0,1).
Then
EX =
∞∑n=1
n(1−p)n−1p = p∞∑
n=1
n(1−p)n−1 = p1
(1− (1−p))2=
1p
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 38: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/38.jpg)
Expectation of Geometric Random VariableTo compute EX for geometric random variable X one need to use quite a nice analytictrick. First we again remind the formula of the sum of geometric progression
∞∑n=0
xn =1
1− x, for x ∈ (0,1).
Then taking the derivatives from both sides we get
ddx
∞∑n=0
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=0
ddx
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=1
nxn−1 =1
(1− x)2, for x ∈ (0,1).
Then
EX =
∞∑n=1
n(1−p)n−1p = p∞∑
n=1
n(1−p)n−1 = p1
(1− (1−p))2=
1p
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 39: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/39.jpg)
Expectation of Geometric Random VariableTo compute EX for geometric random variable X one need to use quite a nice analytictrick. First we again remind the formula of the sum of geometric progression
∞∑n=0
xn =1
1− x, for x ∈ (0,1).
Then taking the derivatives from both sides we get
ddx
∞∑n=0
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=0
ddx
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=1
nxn−1 =1
(1− x)2, for x ∈ (0,1).
Then
EX =
∞∑n=1
n(1−p)n−1p = p∞∑
n=1
n(1−p)n−1 = p1
(1− (1−p))2=
1p
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 40: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/40.jpg)
Expectation of Geometric Random VariableTo compute EX for geometric random variable X one need to use quite a nice analytictrick. First we again remind the formula of the sum of geometric progression
∞∑n=0
xn =1
1− x, for x ∈ (0,1).
Then taking the derivatives from both sides we get
ddx
∞∑n=0
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=0
ddx
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=1
nxn−1 =1
(1− x)2, for x ∈ (0,1).
Then
EX =
∞∑n=1
n(1−p)n−1p = p∞∑
n=1
n(1−p)n−1 = p1
(1− (1−p))2=
1p
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 41: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/41.jpg)
Expectation of Geometric Random VariableTo compute EX for geometric random variable X one need to use quite a nice analytictrick. First we again remind the formula of the sum of geometric progression
∞∑n=0
xn =1
1− x, for x ∈ (0,1).
Then taking the derivatives from both sides we get
ddx
∞∑n=0
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=0
ddx
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=1
nxn−1 =1
(1− x)2, for x ∈ (0,1).
Then
EX =
∞∑n=1
n(1−p)n−1p = p∞∑
n=1
n(1−p)n−1 = p1
(1− (1−p))2=
1p
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 42: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/42.jpg)
Expectation of Geometric Random VariableTo compute EX for geometric random variable X one need to use quite a nice analytictrick. First we again remind the formula of the sum of geometric progression
∞∑n=0
xn =1
1− x, for x ∈ (0,1).
Then taking the derivatives from both sides we get
ddx
∞∑n=0
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=0
ddx
xn =ddx
11− x
, for x ∈ (0,1).
∞∑n=1
nxn−1 =1
(1− x)2, for x ∈ (0,1).
Then
EX =
∞∑n=1
n(1−p)n−1p = p∞∑
n=1
n(1−p)n−1 = p1
(1− (1−p))2=
1p
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 43: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/43.jpg)
Functions of a Random Variable
Random Variable is function from Sample Space to the real line: so you can easilycombine it with other functions from real line to the real line.
Consider a Bernoulli Random Variable X :
P(X = 0) = 1−p and P(X = 1) = p,
and consider a function f (x) = 3x +5, notice that f (0) = 5 and f (1) = 8, so f (X) isa new random variables taking values 5 and 8 such that
P(f (X) = 5) = 1−p and P(f (X) = 8) = p.
BE CAREFULConsider a random variable Y , taking values 0,1,2 such that
P(Y = 0) = .2,P(Y = 1) = .3 and P(Y = 2) = .5.
Consider function g(x) = x2− x +5, then g(0) = 5, g(1) = 5 and g(2) = 7 thusg(Y ) is a new random variable which takes values 5 and 7 and
P(g(Y ) = 5) = P(Y = 0)+P(Y = 1) = .2+ .3= .5,
P(g(Y ) = 7) = .5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 44: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/44.jpg)
Functions of a Random Variable
Random Variable is function from Sample Space to the real line: so you can easilycombine it with other functions from real line to the real line.
Consider a Bernoulli Random Variable X :
P(X = 0) = 1−p and P(X = 1) = p,
and consider a function f (x) = 3x +5,
notice that f (0) = 5 and f (1) = 8, so f (X) isa new random variables taking values 5 and 8 such that
P(f (X) = 5) = 1−p and P(f (X) = 8) = p.
BE CAREFULConsider a random variable Y , taking values 0,1,2 such that
P(Y = 0) = .2,P(Y = 1) = .3 and P(Y = 2) = .5.
Consider function g(x) = x2− x +5, then g(0) = 5, g(1) = 5 and g(2) = 7 thusg(Y ) is a new random variable which takes values 5 and 7 and
P(g(Y ) = 5) = P(Y = 0)+P(Y = 1) = .2+ .3= .5,
P(g(Y ) = 7) = .5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 45: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/45.jpg)
Functions of a Random Variable
Random Variable is function from Sample Space to the real line: so you can easilycombine it with other functions from real line to the real line.
Consider a Bernoulli Random Variable X :
P(X = 0) = 1−p and P(X = 1) = p,
and consider a function f (x) = 3x +5, notice that f (0) = 5 and f (1) = 8,
so f (X) isa new random variables taking values 5 and 8 such that
P(f (X) = 5) = 1−p and P(f (X) = 8) = p.
BE CAREFULConsider a random variable Y , taking values 0,1,2 such that
P(Y = 0) = .2,P(Y = 1) = .3 and P(Y = 2) = .5.
Consider function g(x) = x2− x +5, then g(0) = 5, g(1) = 5 and g(2) = 7 thusg(Y ) is a new random variable which takes values 5 and 7 and
P(g(Y ) = 5) = P(Y = 0)+P(Y = 1) = .2+ .3= .5,
P(g(Y ) = 7) = .5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 46: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/46.jpg)
Functions of a Random Variable
Random Variable is function from Sample Space to the real line: so you can easilycombine it with other functions from real line to the real line.
Consider a Bernoulli Random Variable X :
P(X = 0) = 1−p and P(X = 1) = p,
and consider a function f (x) = 3x +5, notice that f (0) = 5 and f (1) = 8, so f (X) isa new random variables taking values 5 and 8 such that
P(f (X) = 5) = 1−p and P(f (X) = 8) = p.
BE CAREFULConsider a random variable Y , taking values 0,1,2 such that
P(Y = 0) = .2,P(Y = 1) = .3 and P(Y = 2) = .5.
Consider function g(x) = x2− x +5, then g(0) = 5, g(1) = 5 and g(2) = 7 thusg(Y ) is a new random variable which takes values 5 and 7 and
P(g(Y ) = 5) = P(Y = 0)+P(Y = 1) = .2+ .3= .5,
P(g(Y ) = 7) = .5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 47: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/47.jpg)
Functions of a Random Variable
Random Variable is function from Sample Space to the real line: so you can easilycombine it with other functions from real line to the real line.
Consider a Bernoulli Random Variable X :
P(X = 0) = 1−p and P(X = 1) = p,
and consider a function f (x) = 3x +5, notice that f (0) = 5 and f (1) = 8, so f (X) isa new random variables taking values 5 and 8 such that
P(f (X) = 5) = 1−p and P(f (X) = 8) = p.
BE CAREFULConsider a random variable Y , taking values 0,1,2 such that
P(Y = 0) = .2,P(Y = 1) = .3 and P(Y = 2) = .5.
Consider function g(x) = x2− x +5, then g(0) = 5, g(1) = 5 and g(2) = 7 thusg(Y ) is a new random variable which takes values 5 and 7 and
P(g(Y ) = 5) = P(Y = 0)+P(Y = 1) = .2+ .3= .5,
P(g(Y ) = 7) = .5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 48: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/48.jpg)
Functions of a Random Variable
Random Variable is function from Sample Space to the real line: so you can easilycombine it with other functions from real line to the real line.
Consider a Bernoulli Random Variable X :
P(X = 0) = 1−p and P(X = 1) = p,
and consider a function f (x) = 3x +5, notice that f (0) = 5 and f (1) = 8, so f (X) isa new random variables taking values 5 and 8 such that
P(f (X) = 5) = 1−p and P(f (X) = 8) = p.
BE CAREFULConsider a random variable Y , taking values 0,1,2 such that
P(Y = 0) = .2,P(Y = 1) = .3 and P(Y = 2) = .5.
Consider function g(x) = x2− x +5, then g(0) = 5, g(1) = 5 and g(2) = 7 thusg(Y ) is a new random variable which takes values 5 and 7 and
P(g(Y ) = 5) = P(Y = 0)+P(Y = 1) = .2+ .3= .5,
P(g(Y ) = 7) = .5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 49: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/49.jpg)
Functions of a Random Variable
Random Variable is function from Sample Space to the real line: so you can easilycombine it with other functions from real line to the real line.
Consider a Bernoulli Random Variable X :
P(X = 0) = 1−p and P(X = 1) = p,
and consider a function f (x) = 3x +5, notice that f (0) = 5 and f (1) = 8, so f (X) isa new random variables taking values 5 and 8 such that
P(f (X) = 5) = 1−p and P(f (X) = 8) = p.
BE CAREFULConsider a random variable Y , taking values 0,1,2 such that
P(Y = 0) = .2,P(Y = 1) = .3 and P(Y = 2) = .5.
Consider function g(x) = x2− x +5,
then g(0) = 5, g(1) = 5 and g(2) = 7 thusg(Y ) is a new random variable which takes values 5 and 7 and
P(g(Y ) = 5) = P(Y = 0)+P(Y = 1) = .2+ .3= .5,
P(g(Y ) = 7) = .5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 50: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/50.jpg)
Functions of a Random Variable
Random Variable is function from Sample Space to the real line: so you can easilycombine it with other functions from real line to the real line.
Consider a Bernoulli Random Variable X :
P(X = 0) = 1−p and P(X = 1) = p,
and consider a function f (x) = 3x +5, notice that f (0) = 5 and f (1) = 8, so f (X) isa new random variables taking values 5 and 8 such that
P(f (X) = 5) = 1−p and P(f (X) = 8) = p.
BE CAREFULConsider a random variable Y , taking values 0,1,2 such that
P(Y = 0) = .2,P(Y = 1) = .3 and P(Y = 2) = .5.
Consider function g(x) = x2− x +5, then g(0) = 5, g(1) = 5 and g(2) = 7
thusg(Y ) is a new random variable which takes values 5 and 7 and
P(g(Y ) = 5) = P(Y = 0)+P(Y = 1) = .2+ .3= .5,
P(g(Y ) = 7) = .5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 51: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/51.jpg)
Functions of a Random Variable
Random Variable is function from Sample Space to the real line: so you can easilycombine it with other functions from real line to the real line.
Consider a Bernoulli Random Variable X :
P(X = 0) = 1−p and P(X = 1) = p,
and consider a function f (x) = 3x +5, notice that f (0) = 5 and f (1) = 8, so f (X) isa new random variables taking values 5 and 8 such that
P(f (X) = 5) = 1−p and P(f (X) = 8) = p.
BE CAREFULConsider a random variable Y , taking values 0,1,2 such that
P(Y = 0) = .2,P(Y = 1) = .3 and P(Y = 2) = .5.
Consider function g(x) = x2− x +5, then g(0) = 5, g(1) = 5 and g(2) = 7 thusg(Y ) is a new random variable which takes values 5 and 7 and
P(g(Y ) = 5) = P(Y = 0)+P(Y = 1) = .2+ .3= .5,
P(g(Y ) = 7) = .5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 52: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/52.jpg)
Functions of a Random Variable
Random Variable is function from Sample Space to the real line: so you can easilycombine it with other functions from real line to the real line.
Consider a Bernoulli Random Variable X :
P(X = 0) = 1−p and P(X = 1) = p,
and consider a function f (x) = 3x +5, notice that f (0) = 5 and f (1) = 8, so f (X) isa new random variables taking values 5 and 8 such that
P(f (X) = 5) = 1−p and P(f (X) = 8) = p.
BE CAREFULConsider a random variable Y , taking values 0,1,2 such that
P(Y = 0) = .2,P(Y = 1) = .3 and P(Y = 2) = .5.
Consider function g(x) = x2− x +5, then g(0) = 5, g(1) = 5 and g(2) = 7 thusg(Y ) is a new random variable which takes values 5 and 7 and
P(g(Y ) = 5) = P(Y = 0)+P(Y = 1) = .2+ .3= .5,
P(g(Y ) = 7) = .5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 53: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/53.jpg)
Functions of a Random Variable
Random Variable is function from Sample Space to the real line: so you can easilycombine it with other functions from real line to the real line.
Consider a Bernoulli Random Variable X :
P(X = 0) = 1−p and P(X = 1) = p,
and consider a function f (x) = 3x +5, notice that f (0) = 5 and f (1) = 8, so f (X) isa new random variables taking values 5 and 8 such that
P(f (X) = 5) = 1−p and P(f (X) = 8) = p.
BE CAREFULConsider a random variable Y , taking values 0,1,2 such that
P(Y = 0) = .2,P(Y = 1) = .3 and P(Y = 2) = .5.
Consider function g(x) = x2− x +5, then g(0) = 5, g(1) = 5 and g(2) = 7 thusg(Y ) is a new random variable which takes values 5 and 7 and
P(g(Y ) = 5) = P(Y = 0)+P(Y = 1) = .2+ .3= .5,
P(g(Y ) = 7) = .5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 54: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/54.jpg)
Expectation of Functions of a Random Variable
Theorem:Let X be a discrete random variable and a,b are constants, then
E(aX +b) = aE(X)+b
Proof:
E(aX +b)=∑
x :p(x)>0
(axi +b)p(xi )= a
∑x :p(x)>0
xi p(xi )
+b∑
x :p(x)>0
p(xi )= aE(X)+b,
where we used that∑
x :p(x)>0p(xi ) = 1.
Find EY if Y = 3X−7, where X is a binomial random variables with parameters (5, .3)
Solution: Using the formula for expectation of Binomial random variable we getEX = 1.5 and thus
EY = E(3X −7) = 3EX −7= 3∗1.5−7=−2.5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 55: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/55.jpg)
Expectation of Functions of a Random Variable
Theorem:Let X be a discrete random variable and a,b are constants, then
E(aX +b) = aE(X)+b
Proof:
E(aX +b)=∑
x :p(x)>0
(axi +b)p(xi )= a
∑x :p(x)>0
xi p(xi )
+b∑
x :p(x)>0
p(xi )= aE(X)+b,
where we used that∑
x :p(x)>0p(xi ) = 1.
Find EY if Y = 3X−7, where X is a binomial random variables with parameters (5, .3)
Solution: Using the formula for expectation of Binomial random variable we getEX = 1.5 and thus
EY = E(3X −7) = 3EX −7= 3∗1.5−7=−2.5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 56: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/56.jpg)
Expectation of Functions of a Random Variable
Theorem:Let X be a discrete random variable and a,b are constants, then
E(aX +b) = aE(X)+b
Proof:
E(aX +b)=∑
x :p(x)>0
(axi +b)p(xi )= a
∑x :p(x)>0
xi p(xi )
+b∑
x :p(x)>0
p(xi )= aE(X)+b,
where we used that∑
x :p(x)>0p(xi ) = 1.
Find EY if Y = 3X−7, where X is a binomial random variables with parameters (5, .3)
Solution: Using the formula for expectation of Binomial random variable we getEX = 1.5 and thus
EY = E(3X −7) = 3EX −7= 3∗1.5−7=−2.5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.
![Page 57: Lecture 3.2, MATH-57091 Probability and Statistics for ...zvavitch/57091_Lecture_3_2_2014.pdfLecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers. ArtemZvavitch](https://reader033.vdocuments.us/reader033/viewer/2022050106/5f44004f909db040ce1bf735/html5/thumbnails/57.jpg)
Expectation of Functions of a Random Variable
Theorem:Let X be a discrete random variable and a,b are constants, then
E(aX +b) = aE(X)+b
Proof:
E(aX +b)=∑
x :p(x)>0
(axi +b)p(xi )= a
∑x :p(x)>0
xi p(xi )
+b∑
x :p(x)>0
p(xi )= aE(X)+b,
where we used that∑
x :p(x)>0p(xi ) = 1.
Find EY if Y = 3X−7, where X is a binomial random variables with parameters (5, .3)
Solution: Using the formula for expectation of Binomial random variable we getEX = 1.5 and thus
EY = E(3X −7) = 3EX −7= 3∗1.5−7=−2.5.
Artem Zvavitch Lecture 3.2, MATH-57091 Probability and Statistics for High-School Teachers.