07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020...
Transcript of 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020...
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07: Variance, Bernoulli, BinomialLisa YanApril 20, 2020
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Lisa Yan, CS109, 2020
Quick slide reference
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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
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07a_variance_i
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Lisa Yan, CS109, 2020
Stanford, CA𝐸 high = 68°F𝐸 low = 52°F
Washington, DC𝐸 high = 67°F𝐸 low = 51°F
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Average annual weather
Is 𝐸 𝑋 enough?
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Lisa Yan, CS109, 2020
Stanford, CA𝐸 high = 68°F𝐸 low = 52°F
Washington, DC𝐸 high = 67°F𝐸 low = 51°F
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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.
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Lisa Yan, CS109, 2020
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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Lisa Yan, CS109, 2020
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, CS109, 2020
Variance of Stanford weather
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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 124 (°F)2
𝐸 𝑋 = 𝜇 = 68
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
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Lisa Yan, CS109, 2020
Stanford, CA𝐸 high = 68°F
Washington, DC𝐸 high = 67°F
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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
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Properties of Variance
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07b_variance_ii
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Lisa Yan, CS109, 2020
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
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Lisa Yan, CS109, 2020
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
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Lisa Yan, CS109, 2020
Computing variance, a proof
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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, CS109, 2020
Let Y = outcome of a single die roll. Recall 𝐸 𝑌 = 7/2 .Calculate the variance of Y.
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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, CS109, 2020
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
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Lisa Yan, CS109, 2020
Property 2: A proofProperty 2 Var 𝑎𝑋 + 𝑏 = 𝑎!Var 𝑋
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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, CS109, 2020
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, CS109, 2020
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, CS109, 2020
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, CS109, 2020
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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Binomial RV
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07d_binomial
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Lisa Yan, CS109, 2020
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, CS109, 2020 24
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Lisa Yan, CS109, 2020
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)
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Lisa Yan, CS109, 2020
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
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Lisa Yan, CS109, 2020
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:
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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, CS109, 2020
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:
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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
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Lisa Yan, CS109, 2020
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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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 "#!
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Lisa Yan, CS109, 2020
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)
+
+
+
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Lisa Yan, CS109, 2020
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:
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Lisa Yan, CS109, 2020
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 · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/33.jpg)
Lisa Yan, CS109, 2020
No, give me the variance proof right now
33
proofwiki.org
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(live)07: Variance, Bernoulli, and BinomialLisa YanApril 20, 2020
34
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Lisa Yan, CS109, 2020
Reminders: Lecture with• Turn on your camera if you are able, mute your mic in the big room• Virtual backgrounds are encouraged (classroom-appropriate)
Breakout Rooms for meeting your classmates◦ Just like sitting next to someone new◦ This experience is optional: You should be comfortable leaving the room at any time.
We will use Ed instead of Zoom chat
35
Today’s discussion thread: https://us.edstem.org/courses/109/discussion/39075
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Lisa Yan, CS109, 2020
Our first common RVs
36
𝑋 ~ Ber(𝑝)
𝑌 ~ Bin(𝑛, 𝑝)
1. The random variable
2. is distributed as a
3. Bernoulli 4. with parameter
Example: Heads in one coin flip,P(heads) = 0.8 = p
Example: # heads in 40 coin flips,P(heads) = 0.8 = p
otherwise Identify PMF, oridentify as a function of an existing random variable
Review
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Breakout Rooms
Check out the questions on the next slide (Slide 38). Post any clarifications here!
https://us.edstem.org/courses/109/discussion/39075
Breakout rooms: 5 min. Introduce yourself!
37
🤔
![Page 38: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/38.jpg)
Lisa Yan, CS109, 2020
Statistics: Expectation and variance1. a. Let 𝑋 = the outcome of a 4-sided
die roll. What is 𝐸 𝑋 ?b. Let 𝑌 = the sum of three rolls of a
4-sided die. What is 𝐸 𝑌 ?2. a. Let 𝑍 = # of tails on 10 flips of a
biased coin (w.p. 0.4 of heads). What is 𝐸 𝑍 ?
b. What is Var(𝑍)?
3. Compare the variances of𝐵2~Ber 0.1 and 𝐵!~Ber(0.5).
38
🤔
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Lisa Yan, CS109, 2020
Statistics: Expectation and variance1. a. Let 𝑋 = the outcome of a 4-sided
die roll. What is 𝐸 𝑋 ?b. Let 𝑌 = the sum of three rolls of a
4-sided die. What is 𝐸 𝑌 ?2. a. Let 𝑍 = # of tails on 10 flips of a
biased coin (w.p. 0.3 of heads). What is 𝐸 𝑍 ?
b. What is Var(𝑍)?
3. Compare the variances of𝐵2~Ber 0.1 and 𝐵!~Ber(0.5).
39
If you can identify common RVs, just look up statistics instead of re-deriving from definitions.
![Page 40: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/40.jpg)
ThinkSlide 41 has a matching question to go over by yourself. We’ll go over it together afterwards.
Post any clarifications here!https://us.edstem.org/courses/109/discussion/39075
Think by yourself: 2 min
40
🤔(by yourself)
![Page 41: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/41.jpg)
Lisa Yan, CS109, 2020
Visualizing Binomial PMFs
41
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑖 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
𝐸 𝑋 = 𝑛𝑝
C. D.Match the distribution to the graph:1. Bin 10,0.52. Bin 10,0.33. Bin 10,0.74. Bin 5,0.5
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
A. B.
𝑘𝑃𝑋=𝑘
𝑘
𝑃𝑋=𝑘
𝑘
𝑃𝑋=𝑘
𝑘
𝑃𝑋=𝑘
🤔(by yourself)
![Page 42: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/42.jpg)
Lisa Yan, CS109, 2020
Visualizing Binomial PMFs
42
Match the distribution to the graph:1. Bin 10,0.52. Bin 10,0.33. Bin 10,0.74. Bin 5,0.5
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
C. D.
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
A. B.
𝑘𝑃𝑋=𝑘
𝑘
𝑃𝑋=𝑘
𝑘
𝑃𝑋=𝑘
𝑘
𝑃𝑋=𝑘
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑖 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
𝐸 𝑋 = 𝑛𝑝
![Page 43: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/43.jpg)
Lisa Yan, CS109, 2020
Binomial RV is sum of Bernoulli RVs
Bernoulli• 𝑋~Ber(𝑝)
Binomial• 𝑌~Bin 𝑛, 𝑝• The sum of 𝑛 independent
Bernoulli RVs
43
𝑌 =++)&
*
𝑋+ , 𝑋+ ~Ber(𝑝)
+
+
+
Review
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Lisa Yan, CS109, 2020
Galton Board
44
0 1 2 3 4 5
http://web.stanford.edu/class/cs109/demos/galton.html
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ThinkSlide 46 has a question to go over by yourself.
Post any clarifications here!https://us.edstem.org/courses/109/discussion/39075
Think by yourself: 2 min
45
🤔(by yourself)
![Page 46: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/46.jpg)
Lisa Yan, CS109, 2020
Galton Board
46
When a marble hits a pin, it has an equal chance of going left or right.Let 𝐵 = the bucket index a ball drops into.What is the distribution of 𝐵?
0 1 2 3 4 5
𝑛 = 5
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑘 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
🤔(by yourself)
(Interpret: If 𝐵 is a common random variable, report it,
otherwise report PMF)
![Page 47: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/47.jpg)
Lisa Yan, CS109, 2020
Galton Board
47
When a marble hits a pin, it has an equal chance of going left or right.Let 𝐵 = the bucket index a ball drops into.What is the distribution of 𝐵?
0 1 2 3 4 5
𝑛 = 5
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑘 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
• Each pin is an independent trial• One decision made for level 𝑖 = 1, 2, . . , 5• Consider a Bernoulli RV with success 𝑅+ if
ball went right on level 𝑖• Bucket index 𝐵 = # times ball went right
𝐵~Bin(𝑛 = 5, 𝑝 = 0.5)
![Page 48: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/48.jpg)
Lisa Yan, CS109, 2020
Galton Board
48
When a marble hits a pin, it has an equal chance of going left or right.Let 𝐵 = the bucket index a ball drops into.𝐵 is distributed as a Binomial RV,
𝐵~Bin(𝑛 = 5, 𝑝 = 0.5)
0 1 2 3 4 5
𝑛 = 5 Calculate the probability of a ball landing in bucket 𝑘.
𝑃 𝐵 = 0 = 50 0.51 ≈ 0.03
𝑃 𝐵 = 1 = 51 0.51 ≈ 0.16
𝑃 𝐵 = 2 = 52 0.51 ≈ 0.31
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑘 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
![Page 49: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/49.jpg)
Lisa Yan, CS109, 2020
Galton Board
49
Let 𝐵 = the bucket index a ball drops into.𝐵 is distributed as a Binomial RV,
𝐵~Bin(𝑛 = 5, 𝑝 = 0.5)
0 1 2 3 4 5
𝑛 = 5 Calculate the probability of a ball landing in bucket 𝑘.
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑘 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
PMF of Binomial RV!
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Interlude for jokes/announcements
50
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Lisa Yan, CS109, 2020
Announcements
51
JOKE HERE
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Lisa Yan, CS109, 2020
Interesting probability news
52
TikTokRecommendation Algorithm Optimizations
https://www.lesswrong.com/posts/sXqAFdf3a5sA3HMCH/tiktok-recommendation-algorithm-optimizations
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Lisa Yan, CS109, 2020
NBA Finals (RIP) and genetics
53
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Think, thenBreakout Rooms
Check out the questions on the next slide (Slide 55). Post any clarifications here!
https://us.edstem.org/courses/109/discussion/39075
By yourself: 2 min
Breakout rooms: 5 min.
54
🤔
![Page 55: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/55.jpg)
Lisa Yan, CS109, 2020
NBA Finals and genetics1. The Golden State Warriors are going to play the Toronto Raptors in a
7-game series during the 2019 NBA finals.• The Warriors have a probability of 58% of winning each game, independently.• A team wins the series if they win at least 4 games (we play all 7 games).
What is P(Warriors winning)?2. Each person has 2 genes per trait (e.g., eye color).• Child receives 1 gene (equally likely) from each parent• Brown is “dominant”, blue is ”recessive”:• Child has brown eyes if either (or both) genes are brown• Blue eyes only if both genes are blue.
• Parents each have 1 brown and 1 blue gene.A family has 4 children. What is P(3 children with brown eyes)?
55
🤔
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Lisa Yan, CS109, 2020
NBA FinalsThe Golden State Warriors are going to play the TorontoRaptors in a 7-game series during the 2019 NBA finals.• The Warriors have a probability of 58% of
winning each game, independently.• A team wins the series if they win at least 4 games
(we play all 7 games).
What is P(Warriors winning)?
56
1. Define events/ RVs & state goal
𝑋: # games Warriors win𝑋~Bin(7, 0.58)
Want:
Desired probability? (select all that apply)A. 𝑃 𝑋 > 4B. 𝑃 𝑋 ≥ 4C. 𝑃 𝑋 > 3D. 1 − 𝑃 𝑋 ≤ 3E. 1 − 𝑃 𝑋 < 3
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑘 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
![Page 57: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/57.jpg)
Lisa Yan, CS109, 2020
Desired probability? (select all that apply)A. 𝑃 𝑋 > 4B. 𝑃 𝑋 ≥ 4C. 𝑃 𝑋 > 3D. 1 − 𝑃 𝑋 ≤ 3E. 1 − 𝑃 𝑋 < 3
NBA FinalsThe Golden State Warriors are going to play the TorontoRaptors in a 7-game series during the 2019 NBA finals.• The Warriors have a probability of 58% of
winning each game, independently.• A team wins the series if they win at least 4 games
(we play all 7 games).
What is P(Warriors winning)?
57
1. Define events/ RVs & state goal
𝑋: # games Warriors win𝑋~Bin(7, 0.58)
Want:
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑘 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
![Page 58: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/58.jpg)
Lisa Yan, CS109, 2020
NBA FinalsThe Golden State Warriors are going to play the Toronto Raptors in a 7-game series during the 2019 NBA finals.• The Warriors have a probability of 58% of
winning each game, independently.• A team wins the series if they win at least 4 games
(we play all 7 games).
What is P(Warriors winning)?
58
Cool Algebra/Probability Fact: this is identical to the probability of winning if we define winning = first to win 4 games
1. Define events/ RVs & state goal
2. Solve
𝑋: # games Warriors win𝑋~Bin(7, 0.58)
Want: 𝑃 𝑋 ≥ 4
𝑃 𝑋 ≥ 4 = +()2
3
𝑃 𝑋 = 𝑘 = +()2
37𝑘 0.58( 0.42 34(
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑘 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
![Page 59: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/59.jpg)
Lisa Yan, CS109, 2020
Genetic inheritanceEach person has 2 genes per trait (e.g., eye color).• Child receives 1 gene (equally likely) from each parent• Brown is “dominant”, blue is ”recessive”:• Child has brown eyes if either (or both) genes are brown• Blue eyes only if both genes are blue.
• Parents each have 1 brown and 1 blue gene.A family has 4 children. What is P(3 children with brown eyes)?
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𝑋~Bin(𝑛, 𝑝) 𝑝 𝑘 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
A. Product of 4 independent eventsB. Probability treeC. Bernoulli, success 𝑝 = 3 children
with brown eyesD. Binomial, 𝑛 = 3 trials, success
𝑝 = brown-eyed childE. Binomial, 𝑛 = 4 trials, success
𝑝 = brown-eyed child
Subset of ideas:
![Page 60: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/60.jpg)
Lisa Yan, CS109, 2020
Each person has 2 genes per trait (e.g., eye color).• Child receives 1 gene (equally likely) from each parent• Brown is “dominant”, blue is ”recessive”:• Child has brown eyes if either (or both) genes are brown• Blue eyes only if both genes are blue.
• Parents each have 1 brown and 1 blue gene.A family has 4 children. What is P(3 children with brown eyes)?
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Genetic inheritance
1. Define events/ RVs & state goal
3. Solve
𝑋: # brown-eyed children,𝑋~Bin(4, 𝑝)
𝑝: 𝑃 brown−eyed child
Want: 𝑃 𝑋 = 3
2. Identify knownprobabilities
𝑋~Bin(𝑛, 𝑝) 𝑝 𝑘 = 𝑛𝑘 𝑝# 1 − 𝑝 $%#
![Page 61: 07: Variance, Bernoulli, Binomial · 2020. 4. 20. · Bernoulli, Binomial Lisa Yan April 20, 2020 1. Lisa Yan, CS109, 2020 Quick slide reference 2 3 Variance 07a_variance_i 10 Properties](https://reader033.fdocuments.in/reader033/viewer/2022052615/607d97a5a5cb89004b0f5cac/html5/thumbnails/61.jpg)
Lisa Yan, CS109, 2020
See you next time
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