w4 probability distributions
TRANSCRIPT
-
8/2/2019 W4 Probability Distributions
1/41
Centre for Computer Technology
ICT114ICT114Mathematics forMathematics for
ComputingComputing
Week : 4Week : 4
Probability DistributionsProbability Distributions
-
8/2/2019 W4 Probability Distributions
2/41
March 20, 2012March 20, 2012
ObjectivesObjectives
Review Week 3Review Week 3Random VariablesRandom Variables
Discrete Probability DistributionDiscrete Probability DistributionContinuous Probability DistributionContinuous Probability DistributionBinomial DistributionBinomial DistributionNormal DistributionNormal DistributionPoisson DistributionPoisson Distribution
-
8/2/2019 W4 Probability Distributions
3/41
March 20, 2012March 20, 2012
Generalized Multiplication RuleGeneralized Multiplication Rule
If an operation can be performed inIf an operation can be performed in nn11waysways, and if for each of these, and if for each of these a seconda second
operationoperation can be performed incan be performed in nn22 waysways,,andand for each of the first two a thirdfor each of the first two a thirdoperationoperation can be performed incan be performed in nn33 waysways,,
and so forth,and so forth, then the sequence of kthen the sequence of koperations can be performed inoperations can be performed innn11.n.n22.n.n33nnkk ways.ways.
-
8/2/2019 W4 Probability Distributions
4/41
March 20, 2012March 20, 2012
PermutationsPermutations
The number of permutations for n distinctThe number of permutations for n distinctobjects taken r at a timeobjects taken r at a time
n.(n-1).(n-2)(n-r+1)n.(n-1).(n-2)(n-r+1)
The above product is represented byThe above product is represented bynnpprr = n! / (n-r)!= n! / (n-r)!
-
8/2/2019 W4 Probability Distributions
5/41
March 20, 2012March 20, 2012
CombinationsCombinations
The number of combinations ofThe number of combinations ofn distinctn distinctobjects taken r at a time isobjects taken r at a time is
n!n!
nnccrr = --------------= --------------
r! (n-r)!r! (n-r)!
-
8/2/2019 W4 Probability Distributions
6/41
March 20, 2012March 20, 2012
Addition Theorem of ProbabilityAddition Theorem of Probability
If A and B are any two events, thenIf A and B are any two events, then
P(AUB)=P(A)+P(B)-P(AP(AUB)=P(A)+P(B)-P(A B)B)
-
8/2/2019 W4 Probability Distributions
7/41
March 20, 2012March 20, 2012
Conditional ProbabilityConditional Probability The probability of an eventThe probability of an event B occurringB occurring, when, when anan
event A has already occurredevent A has already occurred is calledis calledconditional probability, represented byconditional probability, represented by P(B/A).P(B/A).
The conditional probability of B, given A,The conditional probability of B, given A,denoted by P(B/A), is defined bydenoted by P(B/A), is defined by
P(AP(A B)B)
P(B/A) = ---------------- if P(A)>0P(B/A) = ---------------- if P(A)>0P(A)P(A)
-
8/2/2019 W4 Probability Distributions
8/41
Centre for Computer Technology
Probability DistributionsProbability Distributions
-
8/2/2019 W4 Probability Distributions
9/41
March 20, 2012March 20, 2012
Random Variables AgainRandom Variables Again
In some experimentsIn some experiments we cannot controlwe cannot controlcertain variablescertain variables that affect the outcome,that affect the outcome,
even though most of the conditions are theeven though most of the conditions are thesamesame. Such experiments are called. Such experiments are calledrandom experiments.random experiments.
Set of all possible outcomes is called aSet of all possible outcomes is called asample space or sample.sample space or sample.
-
8/2/2019 W4 Probability Distributions
10/41
March 20, 2012March 20, 2012
Random Variables AgainRandom Variables Again
Let a number be assigned to all theLet a number be assigned to all theelements of a random sample space.elements of a random sample space.
Then there is a function defined on theThen there is a function defined on thesample space.sample space.
This function is called a randomThis function is called a randomfunction or a stochastic function,function or a stochastic function,denoted by a capital letterdenoted by a capital letterX or YX or Y..
-
8/2/2019 W4 Probability Distributions
11/41
March 20, 2012March 20, 2012
Random VariablesRandom VariablesA random variable generally has someA random variable generally has some
specified physical, geometrical or otherspecified physical, geometrical or otherfunctionfunction..
Random variable with a finite number ofRandom variable with a finite number ofvalues is called avalues is called a discrete randomdiscrete randomvariable.variable.
Random variable with an infinite numberRandom variable with an infinite numberof values is called aof values is called a nondiscrete randomnondiscrete randomvariable.variable.
-
8/2/2019 W4 Probability Distributions
12/41
March 20, 2012March 20, 2012
Discrete Probability DistributionDiscrete Probability Distribution
LetLet XX = discrete random variable= discrete random variable
xx11,x,x22,x,x33,...,... = possible values (arranged)= possible values (arranged)
The probability is given byThe probability is given byP(X=xP(X=xkk) = f(x) = f(xkk)) k = 1, 2, 3,.k = 1, 2, 3,.
The probability function or distribution in theThe probability function or distribution in the
Simplified form isSimplified form is
P(X=x) = f(x)P(X=x) = f(x)
-
8/2/2019 W4 Probability Distributions
13/41
March 20, 2012March 20, 2012
Discrete Probability DistributionDiscrete Probability Distribution
f(x) is a probability function iff(x) is a probability function if
1.1. f(x) 0f(x) 0
2.2. f(x) = 1f(x) = 1
xxsum is taken over all possible values of xsum is taken over all possible values of x
-
8/2/2019 W4 Probability Distributions
14/41
March 20, 2012March 20, 2012
Example : Suppose a coin is tossed twice. Let XExample : Suppose a coin is tossed twice. Let X
represent the number of heads that can come up.represent the number of heads that can come up.With each sample point we can associate aWith each sample point we can associate anumber for X as followsnumber for X as follows
P(HH) = P(HT) = P(TH) = P(TT) = P(HH) = P(HT) = P(TH) = P(TT) =
Sample PointSample Point HHHH HTHT THTH TTTTXX 22 11 11 00
-
8/2/2019 W4 Probability Distributions
15/41
March 20, 2012March 20, 2012
ThenThen
P(X=0) = P(TT) = P(X=0) = P(TT) = P(X=1) = P(HTUTH) = P(X=1) = P(HTUTH) =
P(X=2) = P(HH) = P(X=2) = P(HH) =
Thus the probability function is given byThus the probability function is given by
xx 00 11 22
f(x)f(x) 1/41/4 1/21/2 1/41/4
-
8/2/2019 W4 Probability Distributions
16/41
March 20, 2012March 20, 2012
Distribution function for a DiscreteDistribution function for a Discrete
Random VariableRandom VariableThe distribution function for X can be obtainedThe distribution function for X can be obtainedfrom the probability function for all x in (-, ),from the probability function for all x in (-, ),
00 -
-
8/2/2019 W4 Probability Distributions
17/41
March 20, 2012March 20, 2012
Example : In the previous exampleExample : In the previous example
00 -
-
8/2/2019 W4 Probability Distributions
18/41
March 20, 2012March 20, 2012
Continuous Random VariablesContinuous Random Variables
The function f(x) has the propertiesThe function f(x) has the properties
1. f(x) 01. f(x) 0
2. f(x)dx = 12. f(x)dx = 1
--
-
8/2/2019 W4 Probability Distributions
19/41
March 20, 2012March 20, 2012
Preliminaries : Binomial theoremPreliminaries : Binomial theorem
The Binomial theorem gives,The Binomial theorem gives,
( x + y )( x + y )nn
= x= xnn
++nn
CC 11 xxn-1n-1
yy
++ nn CC 22 xxn-2n-2 yy22 ++ ++ nn CC n-2n-2 xx
22 yyn-2n-2
++ nn CC n-1n-1 xx yyn-1n-1 + y+ y nn
= = nn CC rr xx n-rn-ryyrr , r ranges from 0 to n, r ranges from 0 to n
-
8/2/2019 W4 Probability Distributions
20/41
March 20, 2012March 20, 2012
Binomial DistributionBinomial DistributionConsider an experiment, like tossing aConsider an experiment, like tossing a
coin or choosing a marble from an urncoin or choosing a marble from an urnrepeatedly.repeatedly.
Each toss or selection is called a trialEach toss or selection is called a trial In some casesIn some cases the probability will notthe probability will not
change from one trial to another,change from one trial to another, likelike
tossing a coin or die.tossing a coin or die.Such experiments are calledSuch experiments are called independentindependent
or Bernoulli trials.or Bernoulli trials.
-
8/2/2019 W4 Probability Distributions
21/41
March 20, 2012March 20, 2012
Binomial DistributionBinomial Distribution
let p = probability that an event will happenlet p = probability that an event will happen
q = 1-pq = 1-p
probability that the event will failprobability that the event will failThe probability that the event will happenThe probability that the event will happenexactly x times in n trialsexactly x times in n trials
f(x) = P(X=x) =f(x) = P(X=x) = nnCCxx ppxx qq(n-x)(n-x)
-
8/2/2019 W4 Probability Distributions
22/41
March 20, 2012March 20, 2012
Lets do this togetherLets do this together
What is the probability of getting exactlyWhat is the probability of getting exactly2 heads in 6 tosses of a fair coin?2 heads in 6 tosses of a fair coin?
p = ?p = ?
q = ?q = ?
P(X=x) = ?P(X=x) = ?
-
8/2/2019 W4 Probability Distributions
23/41
March 20, 2012March 20, 2012
Let getting a head be the probability ofLet getting a head be the probability of
success, p = success, p = Then the probability of failure, q = Then the probability of failure, q =
Number of trials, n = 6Number of trials, n = 6Number of favorable outcomes, x = 2Number of favorable outcomes, x = 2
Therefore,Therefore, P(X=2) =P(X=2) = 66CC22 pp22 qq(6-2)(6-2)
== 66CC22 ()()22 ()()44
-
8/2/2019 W4 Probability Distributions
24/41
March 20, 2012March 20, 2012
Properties of a Binomial DistributionProperties of a Binomial Distribution
Mean,Mean, = np = np
Variance,Variance, 22 = np(1-p)= np(1-p)= npq= npq
Standard Deviation,Standard Deviation, = (npq) = (npq)
-
8/2/2019 W4 Probability Distributions
25/41
March 20, 2012March 20, 2012
Lets do this togetherLets do this together
Toss a fair coin 100 times, and count theToss a fair coin 100 times, and count thenumber of heads that appear. Find thenumber of heads that appear. Find the
mean, variance and standard deviation ofmean, variance and standard deviation ofthis experiment.this experiment.
-
8/2/2019 W4 Probability Distributions
26/41
March 20, 2012March 20, 2012
Normal DistributionNormal Distribution
Normal distribution is a continuousNormal distribution is a continuousdistributiondistribution..
The variable may take any value betweenThe variable may take any value betweenminus infinity and plus infinity.minus infinity and plus infinity.Most of the probabilities are concentratedMost of the probabilities are concentrated
within three standard deviations from thewithin three standard deviations from themean.mean.
-
8/2/2019 W4 Probability Distributions
27/41
March 20, 2012March 20, 2012
Normal DistributionNormal DistributionA normal distributionA normal distribution is symmetric aboutis symmetric about
mean.mean.There are a large number of normalThere are a large number of normal
distributions, but the normal distributiondistributions, but the normal distributionwithwith mean, = 0mean, = 0 andand standard deviation,standard deviation, = 1 = 1(called standard normal distribution,(called standard normal distribution,
denoted as z )denoted as z ) is of interest.is of interest.The probability values for this areThe probability values for this are
extensively tabulated,extensively tabulated, called the z tablecalled the z table..
-
8/2/2019 W4 Probability Distributions
28/41
March 20, 2012March 20, 2012
Normal DistributionNormal Distribution
Any variable X following normal distribution withAny variable X following normal distribution withmean and standard deviation mean and standard deviation can be linearlycan be linearlytransformed to the standard normal distributiontransformed to the standard normal distribution
by subtracting mean and dividing by standardby subtracting mean and dividing by standarddeviation.deviation.
Z be the standard variable corresponding to XZ be the standard variable corresponding to X
X X Z = ---------------Z = ---------------
-
8/2/2019 W4 Probability Distributions
29/41
March 20, 2012March 20, 2012
Prob (z is between 0 and 1.0) = 0.3413Prob (z is between 0 and 1.0) = 0.3413
We may write this asWe may write this as
Prob (0
-
8/2/2019 W4 Probability Distributions
30/41
March 20, 2012March 20, 2012
Lets do this togetherLets do this together
Exercise : Marks of an examination followsExercise : Marks of an examination followsnormal distribution with mean 70 andnormal distribution with mean 70 and
standard deviation 10. What percentage ofstandard deviation 10. What percentage ofstudents obtained marks (i) between 70students obtained marks (i) between 70and 85 (ii) between 80 and 90 (iii) moreand 85 (ii) between 80 and 90 (iii) morethan 75 (iv) less than 60 (v) between 65than 75 (iv) less than 60 (v) between 65
and 80?and 80?
-
8/2/2019 W4 Probability Distributions
31/41
March 20, 2012March 20, 2012
Mean = 70Mean = 70Standard Deviation = 10Standard Deviation = 10
Prob(70 X 85)Prob(70 X 85)
= Prob ([70-70]/10 z [85-70]/10)= Prob ([70-70]/10 z [85-70]/10)
= Prob (0 z 1.5)= Prob (0 z 1.5)= 0.4332= 0.4332
-
8/2/2019 W4 Probability Distributions
32/41
March 20, 2012March 20, 2012
Prob(65 X 80)Prob(65 X 80)
= Prob ([65-70]/10 z [80-70]/10)= Prob ([65-70]/10 z [80-70]/10)
= Prob ( - 0.5 z 1.0)= Prob ( - 0.5 z 1.0)= Prob ( - 0.5 z 0 )+ Prob ( 0 z 1.0)= Prob ( - 0.5 z 0 )+ Prob ( 0 z 1.0)
= Prob ( 0 z 0.5 ) + Prob ( 0 z 1.0)= Prob ( 0 z 0.5 ) + Prob ( 0 z 1.0)
= 0.1915 + 0.3413 = 0.5328= 0.1915 + 0.3413 = 0.5328
-
8/2/2019 W4 Probability Distributions
33/41
March 20, 2012March 20, 2012
Normal approximation to BinomialNormal approximation to Binomial
If n is large and if p is not too close toIf n is large and if p is not too close toeither 0 or 1either 0 or 1, the binomial distribution can, the binomial distribution can
bebe approximated by a normal distributionapproximated by a normal distributionusing standardized variable.using standardized variable.Suppose X follow binomial distributionSuppose X follow binomial distribution
(n,p) [ that is with mean np and variance(n,p) [ that is with mean np and variancenp(1-p)]np(1-p)]
-
8/2/2019 W4 Probability Distributions
34/41
March 20, 2012March 20, 2012
Normal approximation to BinomialNormal approximation to Binomial
(X np)(X np)
Z = ----------------Z = ---------------- (np(1-p))(np(1-p))
will approximately follow standard normalwill approximately follow standard normaldistributiondistribution
-
8/2/2019 W4 Probability Distributions
35/41
March 20, 2012March 20, 2012
Lets do this togetherLets do this together
A fair coin is tossed 400 times. What is theA fair coin is tossed 400 times. What is theprobability that the number of heads willprobability that the number of heads will
be more than 215?be more than 215?
-
8/2/2019 W4 Probability Distributions
36/41
March 20, 2012March 20, 2012
Poisson DistributionPoisson Distribution
The probability distribution of the PoissonThe probability distribution of the Poissonrandom variable X, in a given time intervalrandom variable X, in a given time intervalis given byis given by
ee-t-t (t)(t)xx
P(x:t) = -------------------P(x:t) = ------------------- x= 0, 1, 2,.x= 0, 1, 2,.x!x!
where is the average number ofwhere is the average number ofoutcomes per unit timeoutcomes per unit time
-
8/2/2019 W4 Probability Distributions
37/41
March 20, 2012March 20, 2012
Lets do this togetherLets do this together
If the probability that an individual willIf the probability that an individual willsuffer a bad reaction from an injection of asuffer a bad reaction from an injection of a
given serum is 0.001, determine thegiven serum is 0.001, determine theprobability that out of 200 individualsprobability that out of 200 individuals
(a) exactly 3(a) exactly 3
(b) more than 2 individuals(b) more than 2 individualswill suffer a bad reactionwill suffer a bad reaction
-
8/2/2019 W4 Probability Distributions
38/41
March 20, 2012March 20, 2012
Given, t = 0.001Given, t = 0.001
ee-0.001-0.001 (0.001)(0.001)33
(a) P(3:0.001) = -----------------------(a) P(3:0.001) = -----------------------
3!3!
(b) P(x>2) = 1 (P(x=0)+P(x=1)+P(x=2))(b) P(x>2) = 1 (P(x=0)+P(x=1)+P(x=2))
-
8/2/2019 W4 Probability Distributions
39/41
March 20, 2012March 20, 2012
SummarySummary Discrete Probability DistributionDiscrete Probability Distribution
1. f(x) 01. f(x) 0
2. f(x) = 12. f(x) = 1
xx
Continuous Probability DistributionContinuous Probability Distribution1. f(x) 01. f(x) 0
2. f(x)dx = 12. f(x)dx = 1 --
-
8/2/2019 W4 Probability Distributions
40/41
March 20, 2012March 20, 2012
SummarySummary Binomial Distribution, f(x) = P(X=x) =Binomial Distribution, f(x) = P(X=x) = nnCCxx ppx
x qq(n-x)(n-x)
p = probability of success, q = 1-p = probability of failurep = probability of success, q = 1-p = probability of failure
X X Normal Distribution, Z = ---------------Normal Distribution, Z = ---------------
ee-t-t (t)(t)xx
Poisson distribution, P(x:t) = -----------------Poisson distribution, P(x:t) = -----------------x!x! is the average outcomes per unit time is the average outcomes per unit time
-
8/2/2019 W4 Probability Distributions
41/41
March 20, 2012March 20, 2012
ReferenceReference
Spiegel, Schiller, Srinivasan : ProbabilitySpiegel, Schiller, Srinivasan : Probabilityand Statisticsand Statistics
M R Spiegel : Theory and Problems ofM R Spiegel : Theory and Problems ofStatistics, Schaum's Outline Series,Statistics, Schaum's Outline Series,McGraw HillMcGraw Hill
http://mathworld.wolfram.comhttp://mathworld.wolfram.com