![Page 1: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/1.jpg)
07: Variance, Bernoulli, BinomialJerry CainApril 12, 2021
1
![Page 2: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/2.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Quick slide reference
2
3 Variance 07a_variance_i
10 Properties of variance 07b_variance_ii
17 Bernoulli RV 07c_bernoulli
22 Binomial RV 07d_binomial
34 Exercises LIVE
![Page 3: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/3.jpg)
Variance
3
07a_variance_i
![Page 4: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/4.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Stanford, CAπΈ high = 68Β°FπΈ low = 52Β°F
Washington, DCπΈ high = 67Β°FπΈ low = 51Β°F
4
Average annual weather
Is πΈ π enough?
![Page 5: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/5.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Stanford, CAπΈ high = 68Β°FπΈ low = 52Β°F
Washington, DCπΈ high = 67Β°FπΈ low = 51Β°F
5
Average annual weather
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
ππ=π₯
ππ=π₯
Stanford high temps Washington high temps
35 50 65 80 90 35 50 65 80 90
68Β°F 67Β°F
Normalized histograms are approximations of PMFs.
![Page 6: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/6.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Variance = βspreadβConsider the following three distributions (PMFs):
β’ Expectation: πΈ π = 3 for all distributionsβ’ But the βspreadβ in the distributions is different!β’ Variance, Var π : a formal quantification of βspreadβ
6
![Page 7: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/7.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Variance
The variance of a random variable π with mean πΈ π = π is
Var π = πΈ π β π (
β’ Also written as: πΈ π β πΈ π !
β’ Note: Var(X) β₯ 0β’ Other names: 2nd central moment, or square of the standard deviation
Var π
def standard deviation SD π = Var π7
Units of π!
Units of π
![Page 8: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/8.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Variance of Stanford weather
8
Varianceof π
Var π = πΈ π β πΈ π !
Stanford, CAπΈ high = 68Β°FπΈ low = 52Β°F
0
0.1
0.2
0.3
0.4
ππ=π₯
Stanford high temps
35 50 65 80 90
π π β π !
57Β°F 121 (Β°F)2
πΈ π = π = 68Β°F
71Β°F 9 (Β°F)2
75Β°F 49 (Β°F)2
69Β°F 1 (Β°F)2
β¦ β¦
Variance πΈ π β π ! = 39 (Β°F)2
Standard deviation = 6.2Β°F
![Page 9: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/9.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Stanford, CAπΈ high = 68Β°F
Washington, DCπΈ high = 67Β°F
9
Comparing variance
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
ππ=π₯
ππ=π₯
Stanford high temps Washington high temps
35 50 65 80 90 35 50 65 80 90
Var π = 39 Β°F ! Var π = 248 Β°F !
Varianceof π
Var π = πΈ π β πΈ π !
68Β°F 67Β°F
![Page 10: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/10.jpg)
Properties of Variance
10
07b_variance_ii
![Page 11: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/11.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Properties of varianceDefinition Var π = πΈ π β πΈ π !
def standard deviation SD π = Var π
Property 1 Var π = πΈ π! β πΈ π !
Property 2 Var ππ + π = π!Var π
11
Units of π!
Units of π
β’ Property 1 is often easier to compute than the definitionβ’ Unlike expectation, variance is not linear
![Page 12: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/12.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Properties of varianceDefinition Var π = πΈ π β πΈ π !
def standard deviation SD π = Var π
Property 1 Var π = πΈ π! β πΈ π !
Property 2 Var ππ + π = π!Var π
12
Units of π!
Units of π
β’ Property 1 is often easier to compute than the definitionβ’ Unlike expectation, variance is not linear
![Page 13: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/13.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Computing variance, a proof
13
Var π = πΈ π β πΈ π !
= πΈ π! β πΈ π !
= πΈ π! β π!= πΈ π! β 2π! + π!= πΈ π! β 2ππΈ π + π!
= πΈ π β π ! Let πΈ π = π
Everyone, please
welcome the second
moment!
Varianceof π
Var π = πΈ π β πΈ π !
= πΈ π! β πΈ π !
=,"
π₯ β π !π π₯
=,"
π₯! β 2ππ₯ + π! π π₯
=,"
π₯!π π₯ β 2π,"
π₯π π₯ + π!,"
π π₯
β 1
![Page 14: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/14.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Let Y = outcome of a single die roll. Recall πΈ π = 7/2 .Calculate the variance of Y.
14
Variance of a 6-sided dieVarianceof π
Var π = πΈ π β πΈ π !
= πΈ π! β πΈ π !
1. Approach #1: Definition
Var π =161 β
72
!+162 β
72
!
+163 β
72
!+164 β
72
!
+165 β
72
!+166 β
72
!
2. Approach #2: A property
πΈ π! =161! + 2! + 3! + 4! + 5! + 6!
= 91/6
Var π = 91/6 β 7/2 !
= 35/12 = 35/12
2nd moment
![Page 15: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/15.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Properties of varianceDefinition Var π = πΈ π β πΈ π !
def standard deviation SD π = Var π
Property 1 Var π = πΈ π! β πΈ π !
Property 2 Var ππ + π = π!Var π
15
Units of π!
Units of π
β’ Property 1 is often easier to compute than the definitionβ’ Unlike expectation, variance is not linear
![Page 16: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/16.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Property 2: A proofProperty 2 Var ππ + π = π!Var π
16
Var ππ + π= πΈ ππ + π ! β πΈ ππ + π ! Property 1
= πΈ π!π! + 2πππ + π! β ππΈ π + π !
= π!πΈ π! + 2πππΈ π + π! β π! πΈ[π] ! + 2πππΈ[π] + π!
= π!πΈ π! β π! πΈ[π] !
= π! πΈ π! β πΈ[π] !
= π!Var π Property 1
Proof:
Factoring/Linearity of Expectation
![Page 17: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/17.jpg)
Bernoulli RV
17
07c_bernoulli
![Page 18: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/18.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Consider an experiment with two outcomes: βsuccessβ and βfailure.βdef A Bernoulli random variable π maps βsuccessβ to 1 and βfailureβ to 0.
Other names: indicator random variable, boolean random variable
Examples:β’ Coin flipβ’ Random binary digitβ’ Whether a disk drive crashed
Bernoulli Random Variable
18
π π = 1 = π 1 = ππ π = 0 = π 0 = 1 β ππ~Ber(π)
Support: {0,1} VarianceExpectation
PMF
πΈ π = πVar π = π(1 β π)
Remember this nice property of expectation. It will come back!
![Page 19: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/19.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Jacob Bernoulli
19
Jacob Bernoulli (1654-1705), also known as βJamesβ, was a Swiss mathematician
One of many mathematicians in Bernoulli familyThe Bernoulli Random Variable is named for him
My academic great14 grandfather
![Page 20: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/20.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Defining Bernoulli RVs
20
Serve an ad.β’ User clicks w.p. 0.2β’ Ignores otherwise
Let π: 1 if clicked
π~Ber(___)π π = 1 = ___π π = 0 = ___
Roll two dice.β’ Success: roll two 6βsβ’ Failure: anything else
Let π : 1 if success
π~Ber(___)
πΈ π = ___
π~Ber(π) π" 1 = ππ" 0 = 1 β ππΈ π = π
Run a programβ’ Crashes w.p. πβ’ Works w.p. 1 β π
Let π: 1 if crash
π~Ber(π)π π = 1 = ππ π = 0 = 1 β π π€
![Page 21: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/21.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Defining Bernoulli RVs
21
Serve an ad.β’ User clicks w.p. 0.2β’ Ignores otherwise
Let π: 1 if clicked
π~Ber(___)π π = 1 = ___π π = 0 = ___
Roll two dice.β’ Success: roll two 6βsβ’ Failure: anything else
Let π : 1 if success
π~Ber(___)
πΈ π = ___
π~Ber(π) π" 1 = ππ" 0 = 1 β ππΈ π = π
Run a programβ’ Crashes w.p. πβ’ Works w.p. 1 β π
Let π: 1 if crash
π~Ber(π)π π = 1 = ππ π = 0 = 1 β π
![Page 22: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/22.jpg)
Binomial RV
22
07d_binomial
![Page 23: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/23.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Consider an experiment: π independent trials of Ber(π) random variables.def A Binomial random variable π is the number of successes in π trials.
Examples:β’ # heads in n coin flipsβ’ # of 1βs in randomly generated length n bit stringβ’ # of disk drives crashed in 1000 computer cluster
(assuming disks crash independently)
Binomial Random Variable
23
π = 0, 1, β¦ , π:
π π = π = π π = ππ π! 1 β π "#!π~Bin(π, π)
Support: {0,1, β¦ , π}
PMF
πΈ π = ππVar π = ππ(1 β π)Variance
Expectation
![Page 24: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/24.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021 24
![Page 25: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/25.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Reiterating notation
The parameters of a Binomial random variable:β’ π: number of independent trialsβ’ π: probability of success on each trial
25
1. The random variable
2. is distributed as a
3. Binomial 4. with parameters
π ~ Bin(π, π)
![Page 26: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/26.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Reiterating notation
If π is a binomial with parameters π and π, the PMF of π is
26
π ~ Bin(π, π)
π π = π = ππ π( 1 β π )*(
Probability Mass Function for a BinomialProbability that πtakes on the value π
![Page 27: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/27.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Three coin flipsThree fair (βheadsβ with π = 0.5) coins are flipped.β’ π is number of headsβ’ π~Bin 3, 0.5
Compute the following event probabilities:
27
π~Bin(π, π) π π = ππ π# 1 β π $%#
π π = 0
π π = 1
π π = 2
π π = 3
π π = 7P(event)
π€
![Page 28: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/28.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Three coin flipsThree fair (βheadsβ with π = 0.5) coins are flipped.β’ π is number of headsβ’ π~Bin 3, 0.5
Compute the following event probabilities:
28
π~Bin(π, π) π π = ππ π# 1 β π $%#
π π = 0 = π 0 = 30 π$ 1 β π % = &
'
π π = 1
π π = 2
π π = 3
π π = 7
= π 1 = 31 π& 1 β π ! = %
'
= π 2 = 32 π! 1 β π & = %
'
= π 3 = 33 π% 1 β π $ = &
'
= π 7 = 0P(event) PMF
Extra math note:By Binomial Theorem,we can proveβ()$* π π = π = 1
![Page 29: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/29.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Consider an experiment: π independent trials of Ber(π) random variables.def A Binomial random variable π is the number of successes in π trials.
Examples:β’ # heads in n coin flipsβ’ # of 1βs in randomly generated length n bit stringβ’ # of disk drives crashed in 1000 computer cluster
(assuming disks crash independently)
Binomial Random Variable
29
π~Bin(π, π)Range: {0,1, β¦ , π} Variance
Expectation
PMF
πΈ π = ππVar π = ππ(1 β π)
π = 0, 1, β¦ , π:
π π = π = π π = ππ π! 1 β π "#!
![Page 30: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/30.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Ber π = Bin(1, π)
Binomial RV is sum of Bernoulli RVs
Bernoulliβ’ π~Ber(π)
Binomialβ’ π~Bin π, πβ’ The sum of π independent
Bernoulli RVs
30
π =,+)&
*
π+ , π+ ~Ber(π)
+
+
+
![Page 31: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/31.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Consider an experiment: π independent trials of Ber(π) random variables.def A Binomial random variable π is the number of successes in π trials.
Examples:β’ # heads in n coin flipsβ’ # of 1βs in randomly generated length n bit stringβ’ # of disk drives crashed in 1000 computer cluster
(assuming disks crash independently)
Binomial Random Variable
31
π~Bin(π, π)Range: {0,1, β¦ , π}
πΈ π = ππVar π = ππ(1 β π)Variance
Expectation
PMF π = 0, 1, β¦ , π:
π π = π = π π = ππ π! 1 β π "#!
Proof:
![Page 32: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/32.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Consider an experiment: π independent trials of Ber(π) random variables.def A Binomial random variable π is the number of successes in π trials.
Examples:β’ # heads in n coin flipsβ’ # of 1βs in randomly generated length n bit stringβ’ # of disk drives crashed in 1000 computer cluster
(assuming disks crash independently)
Binomial Random Variable
32
π~Bin(π, π)Range: {0,1, β¦ , π}
PMF
πΈ π = ππVar π = ππ(1 β π)
Weβll prove this later in the course
VarianceExpectation
π = 0, 1, β¦ , π:
π π = π = π π = ππ π! 1 β π "#!
![Page 33: 07: Variance, Bernoulli, Binomial - Stanford Universityweb.stanford.edu/class/cs109/lectures/07_bernoulli...Bernoulli, Binomial Lisa Yan April 20, 2020 1 Lisa Yan, CS109, 2020 Quick](https://reader033.vdocument.in/reader033/viewer/2022060717/607d98fd448b2b26c0641a54/html5/thumbnails/33.jpg)
Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
No, give me the variance proof right now
33
proofwiki.org