07: variance, bernoulli, binomial - stanford...
TRANSCRIPT
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07: Variance, Bernoulli, BinomialJerry CainApril 12, 2021
1
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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
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Variance
3
07a_variance_i
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Stanford, CA𝐸 high = 68°F𝐸 low = 52°F
Washington, DC𝐸 high = 67°F𝐸 low = 51°F
4
Average annual weather
Is 𝐸 𝑋 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, CA𝐸 high = 68°F𝐸 low = 52°F
Washington, DC𝐸 high = 67°F𝐸 low = 51°F
5
Average annual weather
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
𝑃𝑋=𝑥
𝑃𝑋=𝑥
Stanford high temps Washington high temps
35 50 65 80 90 35 50 65 80 90
68°F 67°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 = “spread”Consider the following three distributions (PMFs):
• Expectation: 𝐸 𝑋 = 3 for all distributions• But the “spread” in the distributions is different!• Variance, Var 𝑋 : a formal quantification of “spread”
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![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 𝑋 with mean 𝐸 𝑋 = 𝜇 is
Var 𝑋 = 𝐸 𝑋 − 𝜇 (
• Also written as: 𝐸 𝑋 − 𝐸 𝑋 !
• Note: Var(X) ≥ 0• Other names: 2nd central moment, or square of the standard deviation
Var 𝑋
def standard deviation SD 𝑋 = Var 𝑋7
Units of 𝑋!
Units of 𝑋
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Variance of Stanford weather
8
Varianceof 𝑋
Var 𝑋 = 𝐸 𝑋 − 𝐸 𝑋 !
Stanford, CA𝐸 high = 68°F𝐸 low = 52°F
0
0.1
0.2
0.3
0.4
𝑃𝑋=𝑥
Stanford high temps
35 50 65 80 90
𝑋 𝑋 − 𝜇 !
57°F 121 (°F)2
𝐸 𝑋 = 𝜇 = 68°F
71°F 9 (°F)2
75°F 49 (°F)2
69°F 1 (°F)2
… …
Variance 𝐸 𝑋 − 𝜇 ! = 39 (°F)2
Standard deviation = 6.2°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, CA𝐸 high = 68°F
Washington, DC𝐸 high = 67°F
9
Comparing variance
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
𝑃𝑋=𝑥
𝑃𝑋=𝑥
Stanford high temps Washington high temps
35 50 65 80 90 35 50 65 80 90
Var 𝑋 = 39 °F ! Var 𝑋 = 248 °F !
Varianceof 𝑋
Var 𝑋 = 𝐸 𝑋 − 𝐸 𝑋 !
68°F 67°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
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Properties of varianceDefinition Var 𝑋 = 𝐸 𝑋 − 𝐸 𝑋 !
def standard deviation SD 𝑋 = Var 𝑋
Property 1 Var 𝑋 = 𝐸 𝑋! − 𝐸 𝑋 !
Property 2 Var 𝑎𝑋 + 𝑏 = 𝑎!Var 𝑋
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Units of 𝑋!
Units of 𝑋
• Property 1 is often easier to compute than the definition• 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 𝑋 = 𝐸 𝑋 − 𝐸 𝑋 !
def standard deviation SD 𝑋 = Var 𝑋
Property 1 Var 𝑋 = 𝐸 𝑋! − 𝐸 𝑋 !
Property 2 Var 𝑎𝑋 + 𝑏 = 𝑎!Var 𝑋
12
Units of 𝑋!
Units of 𝑋
• Property 1 is often easier to compute than the definition• 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 𝑋 = 𝐸 𝑋 − 𝐸 𝑋 !
= 𝐸 𝑋! − 𝐸 𝑋 !
= 𝐸 𝑋! − 𝜇!= 𝐸 𝑋! − 2𝜇! + 𝜇!= 𝐸 𝑋! − 2𝜇𝐸 𝑋 + 𝜇!
= 𝐸 𝑋 − 𝜇 ! Let 𝐸 𝑋 = 𝜇
Everyone, please
welcome the second
moment!
Varianceof 𝑋
Var 𝑋 = 𝐸 𝑋 − 𝐸 𝑋 !
= 𝐸 𝑋! − 𝐸 𝑋 !
=,"
𝑥 − 𝜇 !𝑝 𝑥
=,"
𝑥! − 2𝜇𝑥 + 𝜇! 𝑝 𝑥
=,"
𝑥!𝑝 𝑥 − 2𝜇,"
𝑥𝑝 𝑥 + 𝜇!,"
𝑝 𝑥
⋅ 1
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Let Y = outcome of a single die roll. Recall 𝐸 𝑌 = 7/2 .Calculate the variance of Y.
14
Variance of a 6-sided dieVarianceof 𝑋
Var 𝑋 = 𝐸 𝑋 − 𝐸 𝑋 !
= 𝐸 𝑋! − 𝐸 𝑋 !
1. Approach #1: Definition
Var 𝑌 =161 −
72
!+162 −
72
!
+163 −
72
!+164 −
72
!
+165 −
72
!+166 −
72
!
2. Approach #2: A property
𝐸 𝑌! =161! + 2! + 3! + 4! + 5! + 6!
= 91/6
Var 𝑌 = 91/6 − 7/2 !
= 35/12 = 35/12
2nd moment
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Properties of varianceDefinition Var 𝑋 = 𝐸 𝑋 − 𝐸 𝑋 !
def standard deviation SD 𝑋 = Var 𝑋
Property 1 Var 𝑋 = 𝐸 𝑋! − 𝐸 𝑋 !
Property 2 Var 𝑎𝑋 + 𝑏 = 𝑎!Var 𝑋
15
Units of 𝑋!
Units of 𝑋
• Property 1 is often easier to compute than the definition• 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 𝑎𝑋 + 𝑏 = 𝑎!Var 𝑋
16
Var 𝑎𝑋 + 𝑏= 𝐸 𝑎𝑋 + 𝑏 ! − 𝐸 𝑎𝑋 + 𝑏 ! Property 1
= 𝐸 𝑎!𝑋! + 2𝑎𝑏𝑋 + 𝑏! − 𝑎𝐸 𝑋 + 𝑏 !
= 𝑎!𝐸 𝑋! + 2𝑎𝑏𝐸 𝑋 + 𝑏! − 𝑎! 𝐸[𝑋] ! + 2𝑎𝑏𝐸[𝑋] + 𝑏!
= 𝑎!𝐸 𝑋! − 𝑎! 𝐸[𝑋] !
= 𝑎! 𝐸 𝑋! − 𝐸[𝑋] !
= 𝑎!Var 𝑋 Property 1
Proof:
Factoring/Linearity of Expectation
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Bernoulli RV
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07c_bernoulli
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Consider an experiment with two outcomes: “success” and “failure.”def A Bernoulli random variable 𝑋 maps “success” to 1 and “failure” to 0.
Other names: indicator random variable, boolean random variable
Examples:• Coin flip• Random binary digit• Whether a disk drive crashed
Bernoulli Random Variable
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𝑃 𝑋 = 1 = 𝑝 1 = 𝑝𝑃 𝑋 = 0 = 𝑝 0 = 1 − 𝑝𝑋~Ber(𝑝)
Support: {0,1} VarianceExpectation
PMF
𝐸 𝑋 = 𝑝Var 𝑋 = 𝑝(1 − 𝑝)
Remember this nice property of expectation. It will come back!
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Jacob Bernoulli
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Jacob Bernoulli (1654-1705), also known as “James”, was a Swiss mathematician
One of many mathematicians in Bernoulli familyThe Bernoulli Random Variable is named for him
My academic great14 grandfather
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Defining Bernoulli RVs
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Serve an ad.• User clicks w.p. 0.2• Ignores otherwise
Let 𝑋: 1 if clicked
𝑋~Ber(___)𝑃 𝑋 = 1 = ___𝑃 𝑋 = 0 = ___
Roll two dice.• Success: roll two 6’s• Failure: anything else
Let 𝑋 : 1 if success
𝑋~Ber(___)
𝐸 𝑋 = ___
𝑋~Ber(𝑝) 𝑝" 1 = 𝑝𝑝" 0 = 1 − 𝑝𝐸 𝑋 = 𝑝
Run a program• Crashes w.p. 𝑝• Works w.p. 1 − 𝑝
Let 𝑋: 1 if crash
𝑋~Ber(𝑝)𝑃 𝑋 = 1 = 𝑝𝑃 𝑋 = 0 = 1 − 𝑝 🤔
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Defining Bernoulli RVs
21
Serve an ad.• User clicks w.p. 0.2• Ignores otherwise
Let 𝑋: 1 if clicked
𝑋~Ber(___)𝑃 𝑋 = 1 = ___𝑃 𝑋 = 0 = ___
Roll two dice.• Success: roll two 6’s• Failure: anything else
Let 𝑋 : 1 if success
𝑋~Ber(___)
𝐸 𝑋 = ___
𝑋~Ber(𝑝) 𝑝" 1 = 𝑝𝑝" 0 = 1 − 𝑝𝐸 𝑋 = 𝑝
Run a program• Crashes w.p. 𝑝• Works w.p. 1 − 𝑝
Let 𝑋: 1 if crash
𝑋~Ber(𝑝)𝑃 𝑋 = 1 = 𝑝𝑃 𝑋 = 0 = 1 − 𝑝
![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
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07d_binomial
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Consider an experiment: 𝑛 independent trials of Ber(𝑝) random variables.def A Binomial random variable 𝑋 is the number of successes in 𝑛 trials.
Examples:• # heads in n coin flips• # of 1’s in randomly generated length n bit string• # of disk drives crashed in 1000 computer cluster
(assuming disks crash independently)
Binomial Random Variable
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𝑘 = 0, 1, … , 𝑛:
𝑃 𝑋 = 𝑘 = 𝑝 𝑘 = 𝑛𝑘 𝑝! 1 − 𝑝 "#!𝑋~Bin(𝑛, 𝑝)
Support: {0,1, … , 𝑛}
PMF
𝐸 𝑋 = 𝑛𝑝Var 𝑋 = 𝑛𝑝(1 − 𝑝)Variance
Expectation
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021 24
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Reiterating notation
The parameters of a Binomial random variable:• 𝑛: number of independent trials• 𝑝: probability of success on each trial
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1. The random variable
2. is distributed as a
3. Binomial 4. with parameters
𝑋 ~ Bin(𝑛, 𝑝)
![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 𝑋 is a binomial with parameters 𝑛 and 𝑝, the PMF of 𝑋 is
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𝑋 ~ Bin(𝑛, 𝑝)
𝑃 𝑋 = 𝑘 = 𝑛𝑘 𝑝( 1 − 𝑝 )*(
Probability Mass Function for a BinomialProbability that 𝑋takes on the value 𝑘
![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 (“heads” with 𝑝 = 0.5) coins are flipped.• 𝑋 is number of heads• 𝑋~Bin 3, 0.5
Compute the following event probabilities:
27
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑘 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
𝑃 𝑋 = 0
𝑃 𝑋 = 1
𝑃 𝑋 = 2
𝑃 𝑋 = 3
𝑃 𝑋 = 7P(event)
🤔
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Lisa Yan, Chris Piech, Mehran Sahami, and Jerry Cain, CS109, Spring 2021
Three coin flipsThree fair (“heads” with 𝑝 = 0.5) coins are flipped.• 𝑋 is number of heads• 𝑋~Bin 3, 0.5
Compute the following event probabilities:
28
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑘 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
𝑃 𝑋 = 0 = 𝑝 0 = 30 𝑝$ 1 − 𝑝 % = &
'
𝑃 𝑋 = 1
𝑃 𝑋 = 2
𝑃 𝑋 = 3
𝑃 𝑋 = 7
= 𝑝 1 = 31 𝑝& 1 − 𝑝 ! = %
'
= 𝑝 2 = 32 𝑝! 1 − 𝑝 & = %
'
= 𝑝 3 = 33 𝑝% 1 − 𝑝 $ = &
'
= 𝑝 7 = 0P(event) PMF
Extra math note:By Binomial Theorem,we can prove∑()$* 𝑃 𝑋 = 𝑘 = 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: 𝑛 independent trials of Ber(𝑝) random variables.def A Binomial random variable 𝑋 is the number of successes in 𝑛 trials.
Examples:• # heads in n coin flips• # of 1’s in randomly generated length n bit string• # of disk drives crashed in 1000 computer cluster
(assuming disks crash independently)
Binomial Random Variable
29
𝑋~Bin(𝑛, 𝑝)Range: {0,1, … , 𝑛} Variance
Expectation
PMF
𝐸 𝑋 = 𝑛𝑝Var 𝑋 = 𝑛𝑝(1 − 𝑝)
𝑘 = 0, 1, … , 𝑛:
𝑃 𝑋 = 𝑘 = 𝑝 𝑘 = 𝑛𝑘 𝑝! 1 − 𝑝 "#!
![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 𝑝 = Bin(1, 𝑝)
Binomial RV is sum of Bernoulli RVs
Bernoulli• 𝑋~Ber(𝑝)
Binomial• 𝑌~Bin 𝑛, 𝑝• The sum of 𝑛 independent
Bernoulli RVs
30
𝑌 =,+)&
*
𝑋+ , 𝑋+ ~Ber(𝑝)
+
+
+
![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: 𝑛 independent trials of Ber(𝑝) random variables.def A Binomial random variable 𝑋 is the number of successes in 𝑛 trials.
Examples:• # heads in n coin flips• # of 1’s in randomly generated length n bit string• # of disk drives crashed in 1000 computer cluster
(assuming disks crash independently)
Binomial Random Variable
31
𝑋~Bin(𝑛, 𝑝)Range: {0,1, … , 𝑛}
𝐸 𝑋 = 𝑛𝑝Var 𝑋 = 𝑛𝑝(1 − 𝑝)Variance
Expectation
PMF 𝑘 = 0, 1, … , 𝑛:
𝑃 𝑋 = 𝑘 = 𝑝 𝑘 = 𝑛𝑘 𝑝! 1 − 𝑝 "#!
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: 𝑛 independent trials of Ber(𝑝) random variables.def A Binomial random variable 𝑋 is the number of successes in 𝑛 trials.
Examples:• # heads in n coin flips• # of 1’s in randomly generated length n bit string• # of disk drives crashed in 1000 computer cluster
(assuming disks crash independently)
Binomial Random Variable
32
𝑋~Bin(𝑛, 𝑝)Range: {0,1, … , 𝑛}
PMF
𝐸 𝑋 = 𝑛𝑝Var 𝑋 = 𝑛𝑝(1 − 𝑝)
We’ll prove this later in the course
VarianceExpectation
𝑘 = 0, 1, … , 𝑛:
𝑃 𝑋 = 𝑘 = 𝑝 𝑘 = 𝑛𝑘 𝑝! 1 − 𝑝 "#!
![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