![Page 1: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/1.jpg)
18.600: Lecture 8
Discrete random variables
Scott Sheffield
MIT
![Page 2: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/2.jpg)
Outline
Defining random variables
Probability mass function and distribution function
Recursions
![Page 3: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/3.jpg)
Outline
Defining random variables
Probability mass function and distribution function
Recursions
![Page 4: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/4.jpg)
Random variables
I A random variable X is a function from the state space to thereal numbers.
I Can interpret X as a quantity whose value depends on theoutcome of an experiment.
I Example: toss n coins (so state space consists of the set of all2n possible coin sequences) and let X be number of heads.
I Question: What is P{X = k} in this case?
I Answer:(nk
)/2n, if k ∈ {0, 1, 2, . . . , n}.
![Page 5: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/5.jpg)
Random variables
I A random variable X is a function from the state space to thereal numbers.
I Can interpret X as a quantity whose value depends on theoutcome of an experiment.
I Example: toss n coins (so state space consists of the set of all2n possible coin sequences) and let X be number of heads.
I Question: What is P{X = k} in this case?
I Answer:(nk
)/2n, if k ∈ {0, 1, 2, . . . , n}.
![Page 6: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/6.jpg)
Random variables
I A random variable X is a function from the state space to thereal numbers.
I Can interpret X as a quantity whose value depends on theoutcome of an experiment.
I Example: toss n coins (so state space consists of the set of all2n possible coin sequences) and let X be number of heads.
I Question: What is P{X = k} in this case?
I Answer:(nk
)/2n, if k ∈ {0, 1, 2, . . . , n}.
![Page 7: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/7.jpg)
Random variables
I A random variable X is a function from the state space to thereal numbers.
I Can interpret X as a quantity whose value depends on theoutcome of an experiment.
I Example: toss n coins (so state space consists of the set of all2n possible coin sequences) and let X be number of heads.
I Question: What is P{X = k} in this case?
I Answer:(nk
)/2n, if k ∈ {0, 1, 2, . . . , n}.
![Page 8: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/8.jpg)
Random variables
I A random variable X is a function from the state space to thereal numbers.
I Can interpret X as a quantity whose value depends on theoutcome of an experiment.
I Example: toss n coins (so state space consists of the set of all2n possible coin sequences) and let X be number of heads.
I Question: What is P{X = k} in this case?
I Answer:(nk
)/2n, if k ∈ {0, 1, 2, . . . , n}.
![Page 9: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/9.jpg)
Independence of multiple events
I In n coin toss example, knowing the values of some cointosses tells us nothing about the others.
I Say E1 . . .En are independent if for each{i1, i2, . . . , ik} ⊂ {1, 2, . . . n} we haveP(Ei1Ei2 . . .Eik ) = P(Ei1)P(Ei2) . . .P(Eik ).
I In other words, the product rule works.
I Independence implies P(E1E2E3|E4E5E6) =P(E1)P(E2)P(E3)P(E4)P(E5)P(E6)
P(E4)P(E5)P(E6)= P(E1E2E3), and other similar
statements.
I Does pairwise independence imply independence?
I No. Consider these three events: first coin heads, second coinheads, odd number heads. Pairwise independent, notindependent.
![Page 10: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/10.jpg)
Independence of multiple events
I In n coin toss example, knowing the values of some cointosses tells us nothing about the others.
I Say E1 . . .En are independent if for each{i1, i2, . . . , ik} ⊂ {1, 2, . . . n} we haveP(Ei1Ei2 . . .Eik ) = P(Ei1)P(Ei2) . . .P(Eik ).
I In other words, the product rule works.
I Independence implies P(E1E2E3|E4E5E6) =P(E1)P(E2)P(E3)P(E4)P(E5)P(E6)
P(E4)P(E5)P(E6)= P(E1E2E3), and other similar
statements.
I Does pairwise independence imply independence?
I No. Consider these three events: first coin heads, second coinheads, odd number heads. Pairwise independent, notindependent.
![Page 11: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/11.jpg)
Independence of multiple events
I In n coin toss example, knowing the values of some cointosses tells us nothing about the others.
I Say E1 . . .En are independent if for each{i1, i2, . . . , ik} ⊂ {1, 2, . . . n} we haveP(Ei1Ei2 . . .Eik ) = P(Ei1)P(Ei2) . . .P(Eik ).
I In other words, the product rule works.
I Independence implies P(E1E2E3|E4E5E6) =P(E1)P(E2)P(E3)P(E4)P(E5)P(E6)
P(E4)P(E5)P(E6)= P(E1E2E3), and other similar
statements.
I Does pairwise independence imply independence?
I No. Consider these three events: first coin heads, second coinheads, odd number heads. Pairwise independent, notindependent.
![Page 12: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/12.jpg)
Independence of multiple events
I In n coin toss example, knowing the values of some cointosses tells us nothing about the others.
I Say E1 . . .En are independent if for each{i1, i2, . . . , ik} ⊂ {1, 2, . . . n} we haveP(Ei1Ei2 . . .Eik ) = P(Ei1)P(Ei2) . . .P(Eik ).
I In other words, the product rule works.
I Independence implies P(E1E2E3|E4E5E6) =P(E1)P(E2)P(E3)P(E4)P(E5)P(E6)
P(E4)P(E5)P(E6)= P(E1E2E3), and other similar
statements.
I Does pairwise independence imply independence?
I No. Consider these three events: first coin heads, second coinheads, odd number heads. Pairwise independent, notindependent.
![Page 13: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/13.jpg)
Independence of multiple events
I In n coin toss example, knowing the values of some cointosses tells us nothing about the others.
I Say E1 . . .En are independent if for each{i1, i2, . . . , ik} ⊂ {1, 2, . . . n} we haveP(Ei1Ei2 . . .Eik ) = P(Ei1)P(Ei2) . . .P(Eik ).
I In other words, the product rule works.
I Independence implies P(E1E2E3|E4E5E6) =P(E1)P(E2)P(E3)P(E4)P(E5)P(E6)
P(E4)P(E5)P(E6)= P(E1E2E3), and other similar
statements.
I Does pairwise independence imply independence?
I No. Consider these three events: first coin heads, second coinheads, odd number heads. Pairwise independent, notindependent.
![Page 14: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/14.jpg)
Independence of multiple events
I In n coin toss example, knowing the values of some cointosses tells us nothing about the others.
I Say E1 . . .En are independent if for each{i1, i2, . . . , ik} ⊂ {1, 2, . . . n} we haveP(Ei1Ei2 . . .Eik ) = P(Ei1)P(Ei2) . . .P(Eik ).
I In other words, the product rule works.
I Independence implies P(E1E2E3|E4E5E6) =P(E1)P(E2)P(E3)P(E4)P(E5)P(E6)
P(E4)P(E5)P(E6)= P(E1E2E3), and other similar
statements.
I Does pairwise independence imply independence?
I No. Consider these three events: first coin heads, second coinheads, odd number heads. Pairwise independent, notindependent.
![Page 15: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/15.jpg)
Examples
I Shuffle n cards, and let X be the position of the jth card.State space consists of all n! possible orderings. X takesvalues in {1, 2, . . . , n} depending on the ordering.
I Question: What is P{X = k} in this case?
I Answer: 1/n, if k ∈ {1, 2, . . . , n}.I Now say we roll three dice and let Y be sum of the values on
the dice. What is P{Y = 5}?I 6/216
![Page 16: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/16.jpg)
Examples
I Shuffle n cards, and let X be the position of the jth card.State space consists of all n! possible orderings. X takesvalues in {1, 2, . . . , n} depending on the ordering.
I Question: What is P{X = k} in this case?
I Answer: 1/n, if k ∈ {1, 2, . . . , n}.I Now say we roll three dice and let Y be sum of the values on
the dice. What is P{Y = 5}?I 6/216
![Page 17: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/17.jpg)
Examples
I Shuffle n cards, and let X be the position of the jth card.State space consists of all n! possible orderings. X takesvalues in {1, 2, . . . , n} depending on the ordering.
I Question: What is P{X = k} in this case?
I Answer: 1/n, if k ∈ {1, 2, . . . , n}.
I Now say we roll three dice and let Y be sum of the values onthe dice. What is P{Y = 5}?
I 6/216
![Page 18: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/18.jpg)
Examples
I Shuffle n cards, and let X be the position of the jth card.State space consists of all n! possible orderings. X takesvalues in {1, 2, . . . , n} depending on the ordering.
I Question: What is P{X = k} in this case?
I Answer: 1/n, if k ∈ {1, 2, . . . , n}.I Now say we roll three dice and let Y be sum of the values on
the dice. What is P{Y = 5}?
I 6/216
![Page 19: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/19.jpg)
Examples
I Shuffle n cards, and let X be the position of the jth card.State space consists of all n! possible orderings. X takesvalues in {1, 2, . . . , n} depending on the ordering.
I Question: What is P{X = k} in this case?
I Answer: 1/n, if k ∈ {1, 2, . . . , n}.I Now say we roll three dice and let Y be sum of the values on
the dice. What is P{Y = 5}?I 6/216
![Page 20: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/20.jpg)
Indicators
I Given any event E , can define an indicator random variable,i.e., let X be random variable equal to 1 on the event E and 0otherwise. Write this as X = 1E .
I The value of 1E (either 1 or 0) indicates whether the eventhas occurred.
I If E1,E2, . . .Ek are events then X =∑k
i=1 1Eiis the number
of these events that occur.
I Example: in n-hat shuffle problem, let Ei be the event ithperson gets own hat.
I Then∑n
i=1 1Eiis total number of people who get own hats.
I Writing random variable as sum of indicators: frequentlyuseful, sometimes confusing.
![Page 21: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/21.jpg)
Indicators
I Given any event E , can define an indicator random variable,i.e., let X be random variable equal to 1 on the event E and 0otherwise. Write this as X = 1E .
I The value of 1E (either 1 or 0) indicates whether the eventhas occurred.
I If E1,E2, . . .Ek are events then X =∑k
i=1 1Eiis the number
of these events that occur.
I Example: in n-hat shuffle problem, let Ei be the event ithperson gets own hat.
I Then∑n
i=1 1Eiis total number of people who get own hats.
I Writing random variable as sum of indicators: frequentlyuseful, sometimes confusing.
![Page 22: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/22.jpg)
Indicators
I Given any event E , can define an indicator random variable,i.e., let X be random variable equal to 1 on the event E and 0otherwise. Write this as X = 1E .
I The value of 1E (either 1 or 0) indicates whether the eventhas occurred.
I If E1,E2, . . .Ek are events then X =∑k
i=1 1Eiis the number
of these events that occur.
I Example: in n-hat shuffle problem, let Ei be the event ithperson gets own hat.
I Then∑n
i=1 1Eiis total number of people who get own hats.
I Writing random variable as sum of indicators: frequentlyuseful, sometimes confusing.
![Page 23: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/23.jpg)
Indicators
I Given any event E , can define an indicator random variable,i.e., let X be random variable equal to 1 on the event E and 0otherwise. Write this as X = 1E .
I The value of 1E (either 1 or 0) indicates whether the eventhas occurred.
I If E1,E2, . . .Ek are events then X =∑k
i=1 1Eiis the number
of these events that occur.
I Example: in n-hat shuffle problem, let Ei be the event ithperson gets own hat.
I Then∑n
i=1 1Eiis total number of people who get own hats.
I Writing random variable as sum of indicators: frequentlyuseful, sometimes confusing.
![Page 24: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/24.jpg)
Indicators
I Given any event E , can define an indicator random variable,i.e., let X be random variable equal to 1 on the event E and 0otherwise. Write this as X = 1E .
I The value of 1E (either 1 or 0) indicates whether the eventhas occurred.
I If E1,E2, . . .Ek are events then X =∑k
i=1 1Eiis the number
of these events that occur.
I Example: in n-hat shuffle problem, let Ei be the event ithperson gets own hat.
I Then∑n
i=1 1Eiis total number of people who get own hats.
I Writing random variable as sum of indicators: frequentlyuseful, sometimes confusing.
![Page 25: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/25.jpg)
Indicators
I Given any event E , can define an indicator random variable,i.e., let X be random variable equal to 1 on the event E and 0otherwise. Write this as X = 1E .
I The value of 1E (either 1 or 0) indicates whether the eventhas occurred.
I If E1,E2, . . .Ek are events then X =∑k
i=1 1Eiis the number
of these events that occur.
I Example: in n-hat shuffle problem, let Ei be the event ithperson gets own hat.
I Then∑n
i=1 1Eiis total number of people who get own hats.
I Writing random variable as sum of indicators: frequentlyuseful, sometimes confusing.
![Page 26: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/26.jpg)
Outline
Defining random variables
Probability mass function and distribution function
Recursions
![Page 27: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/27.jpg)
Outline
Defining random variables
Probability mass function and distribution function
Recursions
![Page 28: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/28.jpg)
Probability mass function
I Say X is a discrete random variable if (with probability one)it takes one of a countable set of values.
I For each a in this countable set, write p(a) := P{X = a}.Call p the probability mass function.
I For the cumulative distribution function, writeF (a) = P{X ≤ a} =
∑x≤a p(x).
I Example: Let T1,T2,T3, . . . be sequence of independent faircoin tosses (each taking values in {H,T}) and let X be thesmallest j for which Tj = H.
I What is p(k) = P{X = k} (for k ∈ Z) in this case?
I p(k) = (1/2)k
I What about FX (k)?
I 1− (1/2)k
![Page 29: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/29.jpg)
Probability mass function
I Say X is a discrete random variable if (with probability one)it takes one of a countable set of values.
I For each a in this countable set, write p(a) := P{X = a}.Call p the probability mass function.
I For the cumulative distribution function, writeF (a) = P{X ≤ a} =
∑x≤a p(x).
I Example: Let T1,T2,T3, . . . be sequence of independent faircoin tosses (each taking values in {H,T}) and let X be thesmallest j for which Tj = H.
I What is p(k) = P{X = k} (for k ∈ Z) in this case?
I p(k) = (1/2)k
I What about FX (k)?
I 1− (1/2)k
![Page 30: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/30.jpg)
Probability mass function
I Say X is a discrete random variable if (with probability one)it takes one of a countable set of values.
I For each a in this countable set, write p(a) := P{X = a}.Call p the probability mass function.
I For the cumulative distribution function, writeF (a) = P{X ≤ a} =
∑x≤a p(x).
I Example: Let T1,T2,T3, . . . be sequence of independent faircoin tosses (each taking values in {H,T}) and let X be thesmallest j for which Tj = H.
I What is p(k) = P{X = k} (for k ∈ Z) in this case?
I p(k) = (1/2)k
I What about FX (k)?
I 1− (1/2)k
![Page 31: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/31.jpg)
Probability mass function
I Say X is a discrete random variable if (with probability one)it takes one of a countable set of values.
I For each a in this countable set, write p(a) := P{X = a}.Call p the probability mass function.
I For the cumulative distribution function, writeF (a) = P{X ≤ a} =
∑x≤a p(x).
I Example: Let T1,T2,T3, . . . be sequence of independent faircoin tosses (each taking values in {H,T}) and let X be thesmallest j for which Tj = H.
I What is p(k) = P{X = k} (for k ∈ Z) in this case?
I p(k) = (1/2)k
I What about FX (k)?
I 1− (1/2)k
![Page 32: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/32.jpg)
Probability mass function
I Say X is a discrete random variable if (with probability one)it takes one of a countable set of values.
I For each a in this countable set, write p(a) := P{X = a}.Call p the probability mass function.
I For the cumulative distribution function, writeF (a) = P{X ≤ a} =
∑x≤a p(x).
I Example: Let T1,T2,T3, . . . be sequence of independent faircoin tosses (each taking values in {H,T}) and let X be thesmallest j for which Tj = H.
I What is p(k) = P{X = k} (for k ∈ Z) in this case?
I p(k) = (1/2)k
I What about FX (k)?
I 1− (1/2)k
![Page 33: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/33.jpg)
Probability mass function
I Say X is a discrete random variable if (with probability one)it takes one of a countable set of values.
I For each a in this countable set, write p(a) := P{X = a}.Call p the probability mass function.
I For the cumulative distribution function, writeF (a) = P{X ≤ a} =
∑x≤a p(x).
I Example: Let T1,T2,T3, . . . be sequence of independent faircoin tosses (each taking values in {H,T}) and let X be thesmallest j for which Tj = H.
I What is p(k) = P{X = k} (for k ∈ Z) in this case?
I p(k) = (1/2)k
I What about FX (k)?
I 1− (1/2)k
![Page 34: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/34.jpg)
Probability mass function
I Say X is a discrete random variable if (with probability one)it takes one of a countable set of values.
I For each a in this countable set, write p(a) := P{X = a}.Call p the probability mass function.
I For the cumulative distribution function, writeF (a) = P{X ≤ a} =
∑x≤a p(x).
I Example: Let T1,T2,T3, . . . be sequence of independent faircoin tosses (each taking values in {H,T}) and let X be thesmallest j for which Tj = H.
I What is p(k) = P{X = k} (for k ∈ Z) in this case?
I p(k) = (1/2)k
I What about FX (k)?
I 1− (1/2)k
![Page 35: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/35.jpg)
Probability mass function
I Say X is a discrete random variable if (with probability one)it takes one of a countable set of values.
I For each a in this countable set, write p(a) := P{X = a}.Call p the probability mass function.
I For the cumulative distribution function, writeF (a) = P{X ≤ a} =
∑x≤a p(x).
I Example: Let T1,T2,T3, . . . be sequence of independent faircoin tosses (each taking values in {H,T}) and let X be thesmallest j for which Tj = H.
I What is p(k) = P{X = k} (for k ∈ Z) in this case?
I p(k) = (1/2)k
I What about FX (k)?
I 1− (1/2)k
![Page 36: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/36.jpg)
Another example
I Another example: let X be non-negative integer such thatp(k) = P{X = k} = e−λλk/k!.
I Recall Taylor expansion∑∞
k=0 λk/k! = eλ.
I In this example, X is called a Poisson random variable withintensity λ.
I Question: what is the state space in this example?
I Answer: Didn’t specify. One possibility would be to definestate space as S = {0, 1, 2, . . .} and define X (as a functionon S) by X (j) = j . The probability function would bedetermined by P(S) =
∑k∈S e
−λλk/k!.
I Are there other choices of S and P — and other functions Xfrom S to P — for which the values of P{X = k} are thesame?
I Yes. “X is a Poisson random variable with intensity λ” isstatement only about the probability mass function of X .
![Page 37: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/37.jpg)
Another example
I Another example: let X be non-negative integer such thatp(k) = P{X = k} = e−λλk/k!.
I Recall Taylor expansion∑∞
k=0 λk/k! = eλ.
I In this example, X is called a Poisson random variable withintensity λ.
I Question: what is the state space in this example?
I Answer: Didn’t specify. One possibility would be to definestate space as S = {0, 1, 2, . . .} and define X (as a functionon S) by X (j) = j . The probability function would bedetermined by P(S) =
∑k∈S e
−λλk/k!.
I Are there other choices of S and P — and other functions Xfrom S to P — for which the values of P{X = k} are thesame?
I Yes. “X is a Poisson random variable with intensity λ” isstatement only about the probability mass function of X .
![Page 38: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/38.jpg)
Another example
I Another example: let X be non-negative integer such thatp(k) = P{X = k} = e−λλk/k!.
I Recall Taylor expansion∑∞
k=0 λk/k! = eλ.
I In this example, X is called a Poisson random variable withintensity λ.
I Question: what is the state space in this example?
I Answer: Didn’t specify. One possibility would be to definestate space as S = {0, 1, 2, . . .} and define X (as a functionon S) by X (j) = j . The probability function would bedetermined by P(S) =
∑k∈S e
−λλk/k!.
I Are there other choices of S and P — and other functions Xfrom S to P — for which the values of P{X = k} are thesame?
I Yes. “X is a Poisson random variable with intensity λ” isstatement only about the probability mass function of X .
![Page 39: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/39.jpg)
Another example
I Another example: let X be non-negative integer such thatp(k) = P{X = k} = e−λλk/k!.
I Recall Taylor expansion∑∞
k=0 λk/k! = eλ.
I In this example, X is called a Poisson random variable withintensity λ.
I Question: what is the state space in this example?
I Answer: Didn’t specify. One possibility would be to definestate space as S = {0, 1, 2, . . .} and define X (as a functionon S) by X (j) = j . The probability function would bedetermined by P(S) =
∑k∈S e
−λλk/k!.
I Are there other choices of S and P — and other functions Xfrom S to P — for which the values of P{X = k} are thesame?
I Yes. “X is a Poisson random variable with intensity λ” isstatement only about the probability mass function of X .
![Page 40: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/40.jpg)
Another example
I Another example: let X be non-negative integer such thatp(k) = P{X = k} = e−λλk/k!.
I Recall Taylor expansion∑∞
k=0 λk/k! = eλ.
I In this example, X is called a Poisson random variable withintensity λ.
I Question: what is the state space in this example?
I Answer: Didn’t specify. One possibility would be to definestate space as S = {0, 1, 2, . . .} and define X (as a functionon S) by X (j) = j . The probability function would bedetermined by P(S) =
∑k∈S e
−λλk/k!.
I Are there other choices of S and P — and other functions Xfrom S to P — for which the values of P{X = k} are thesame?
I Yes. “X is a Poisson random variable with intensity λ” isstatement only about the probability mass function of X .
![Page 41: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/41.jpg)
Another example
I Another example: let X be non-negative integer such thatp(k) = P{X = k} = e−λλk/k!.
I Recall Taylor expansion∑∞
k=0 λk/k! = eλ.
I In this example, X is called a Poisson random variable withintensity λ.
I Question: what is the state space in this example?
I Answer: Didn’t specify. One possibility would be to definestate space as S = {0, 1, 2, . . .} and define X (as a functionon S) by X (j) = j . The probability function would bedetermined by P(S) =
∑k∈S e
−λλk/k!.
I Are there other choices of S and P — and other functions Xfrom S to P — for which the values of P{X = k} are thesame?
I Yes. “X is a Poisson random variable with intensity λ” isstatement only about the probability mass function of X .
![Page 42: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/42.jpg)
Another example
I Another example: let X be non-negative integer such thatp(k) = P{X = k} = e−λλk/k!.
I Recall Taylor expansion∑∞
k=0 λk/k! = eλ.
I In this example, X is called a Poisson random variable withintensity λ.
I Question: what is the state space in this example?
I Answer: Didn’t specify. One possibility would be to definestate space as S = {0, 1, 2, . . .} and define X (as a functionon S) by X (j) = j . The probability function would bedetermined by P(S) =
∑k∈S e
−λλk/k!.
I Are there other choices of S and P — and other functions Xfrom S to P — for which the values of P{X = k} are thesame?
I Yes. “X is a Poisson random variable with intensity λ” isstatement only about the probability mass function of X .
![Page 43: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/43.jpg)
Outline
Defining random variables
Probability mass function and distribution function
Recursions
![Page 44: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/44.jpg)
Outline
Defining random variables
Probability mass function and distribution function
Recursions
![Page 45: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/45.jpg)
Using Bayes’ rule to set up recursions
I Gambler one has positive integer m dollars, gambler two haspositive integer n dollars. Take turns making one dollar betsuntil one runs out of money. What is probability first gamblerruns out of money first?
I n/(m + n)
I Gambler’s ruin: what if gambler one has an unlimitedamount of money?
I Wins eventually with probability one.
I Problem of points: in sequence of independent fair cointosses, what is probability Pn,m to see n heads before seeing mtails?
I Observe: Pn,m is equivalent to the probability of having n ormore heads in first m + n − 1 trials.
I Probability of exactly n heads in m + n − 1 trials is(m+n−1
n
).
I Famous correspondence by Fermat and Pascal. Led Pascal towrite Le Triangle Arithmetique.
![Page 46: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/46.jpg)
Using Bayes’ rule to set up recursions
I Gambler one has positive integer m dollars, gambler two haspositive integer n dollars. Take turns making one dollar betsuntil one runs out of money. What is probability first gamblerruns out of money first?
I n/(m + n)
I Gambler’s ruin: what if gambler one has an unlimitedamount of money?
I Wins eventually with probability one.
I Problem of points: in sequence of independent fair cointosses, what is probability Pn,m to see n heads before seeing mtails?
I Observe: Pn,m is equivalent to the probability of having n ormore heads in first m + n − 1 trials.
I Probability of exactly n heads in m + n − 1 trials is(m+n−1
n
).
I Famous correspondence by Fermat and Pascal. Led Pascal towrite Le Triangle Arithmetique.
![Page 47: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/47.jpg)
Using Bayes’ rule to set up recursions
I Gambler one has positive integer m dollars, gambler two haspositive integer n dollars. Take turns making one dollar betsuntil one runs out of money. What is probability first gamblerruns out of money first?
I n/(m + n)
I Gambler’s ruin: what if gambler one has an unlimitedamount of money?
I Wins eventually with probability one.
I Problem of points: in sequence of independent fair cointosses, what is probability Pn,m to see n heads before seeing mtails?
I Observe: Pn,m is equivalent to the probability of having n ormore heads in first m + n − 1 trials.
I Probability of exactly n heads in m + n − 1 trials is(m+n−1
n
).
I Famous correspondence by Fermat and Pascal. Led Pascal towrite Le Triangle Arithmetique.
![Page 48: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/48.jpg)
Using Bayes’ rule to set up recursions
I Gambler one has positive integer m dollars, gambler two haspositive integer n dollars. Take turns making one dollar betsuntil one runs out of money. What is probability first gamblerruns out of money first?
I n/(m + n)
I Gambler’s ruin: what if gambler one has an unlimitedamount of money?
I Wins eventually with probability one.
I Problem of points: in sequence of independent fair cointosses, what is probability Pn,m to see n heads before seeing mtails?
I Observe: Pn,m is equivalent to the probability of having n ormore heads in first m + n − 1 trials.
I Probability of exactly n heads in m + n − 1 trials is(m+n−1
n
).
I Famous correspondence by Fermat and Pascal. Led Pascal towrite Le Triangle Arithmetique.
![Page 49: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/49.jpg)
Using Bayes’ rule to set up recursions
I Gambler one has positive integer m dollars, gambler two haspositive integer n dollars. Take turns making one dollar betsuntil one runs out of money. What is probability first gamblerruns out of money first?
I n/(m + n)
I Gambler’s ruin: what if gambler one has an unlimitedamount of money?
I Wins eventually with probability one.
I Problem of points: in sequence of independent fair cointosses, what is probability Pn,m to see n heads before seeing mtails?
I Observe: Pn,m is equivalent to the probability of having n ormore heads in first m + n − 1 trials.
I Probability of exactly n heads in m + n − 1 trials is(m+n−1
n
).
I Famous correspondence by Fermat and Pascal. Led Pascal towrite Le Triangle Arithmetique.
![Page 50: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/50.jpg)
Using Bayes’ rule to set up recursions
I Gambler one has positive integer m dollars, gambler two haspositive integer n dollars. Take turns making one dollar betsuntil one runs out of money. What is probability first gamblerruns out of money first?
I n/(m + n)
I Gambler’s ruin: what if gambler one has an unlimitedamount of money?
I Wins eventually with probability one.
I Problem of points: in sequence of independent fair cointosses, what is probability Pn,m to see n heads before seeing mtails?
I Observe: Pn,m is equivalent to the probability of having n ormore heads in first m + n − 1 trials.
I Probability of exactly n heads in m + n − 1 trials is(m+n−1
n
).
I Famous correspondence by Fermat and Pascal. Led Pascal towrite Le Triangle Arithmetique.
![Page 51: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/51.jpg)
Using Bayes’ rule to set up recursions
I Gambler one has positive integer m dollars, gambler two haspositive integer n dollars. Take turns making one dollar betsuntil one runs out of money. What is probability first gamblerruns out of money first?
I n/(m + n)
I Gambler’s ruin: what if gambler one has an unlimitedamount of money?
I Wins eventually with probability one.
I Problem of points: in sequence of independent fair cointosses, what is probability Pn,m to see n heads before seeing mtails?
I Observe: Pn,m is equivalent to the probability of having n ormore heads in first m + n − 1 trials.
I Probability of exactly n heads in m + n − 1 trials is(m+n−1
n
).
I Famous correspondence by Fermat and Pascal. Led Pascal towrite Le Triangle Arithmetique.
![Page 52: 18.600: Lecture 8 .1in Discrete random variablesmath.mit.edu/~sheffield/2019600/Lecture8.pdf · Random variables I A random variable X is a function from the state space to the real](https://reader034.vdocuments.us/reader034/viewer/2022042314/5f02efb07e708231d406be94/html5/thumbnails/52.jpg)
Using Bayes’ rule to set up recursions
I Gambler one has positive integer m dollars, gambler two haspositive integer n dollars. Take turns making one dollar betsuntil one runs out of money. What is probability first gamblerruns out of money first?
I n/(m + n)
I Gambler’s ruin: what if gambler one has an unlimitedamount of money?
I Wins eventually with probability one.
I Problem of points: in sequence of independent fair cointosses, what is probability Pn,m to see n heads before seeing mtails?
I Observe: Pn,m is equivalent to the probability of having n ormore heads in first m + n − 1 trials.
I Probability of exactly n heads in m + n − 1 trials is(m+n−1
n
).
I Famous correspondence by Fermat and Pascal. Led Pascal towrite Le Triangle Arithmetique.