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" " " EAT I RHI Efx I EH ] a ] ) ( ftp.b.PEF-bD Efx - - PowerPoint PPT Presentation

Sep 30, 2022 234 likes •296 views

Last Time... Product of RVs Let X be a RV with values in A . I Variance measures deviation from mean Let Y be a RV with values in B . Covariance and Correlation I Variance is additive for independent RVs What does the distribution of XY look like?

slide-1
SLIDE 1

Covariance and Correlation

CS 70, Summer 2019 Lecture 22, 7/31/19

1 / 27

Last Time...

I Variance measures deviation from mean I Variance is additive for independent RVs I Use linearity of expectation and indicator variables Today: I Proof that var. is additive for ind. RVs I Talk about covariance and correlation I Some RV practice (if time)

2 / 27

Product of RVs

Let X be a RV with values in A. Let Y be a RV with values in B. What does the distribution of XY look like? What if X, Y are independent?

3 / 27

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For Independent RVs...

If X and Y are independent, we can show that: E[XY ] = E[X] · E[Y ]

4 / 27

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Variance is Additive for Ind. RVs

If X and Y are independent, we can show that: Var(X + Y ) = Var[X] + Var[Y ]

5 / 27

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Can You Multiply?

If X and Y are independent, then: E[XY ] = Is the converse true?

6 / 27

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slide-2
SLIDE 2

Covariance

Measures “how independent” two RVs are. Let E[X] = µ1, and let E[Y ] = µ2. Cov(X, Y ) = Alternate Form: Cov(X, Y ) =

7 / 27

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Example: Coin Flips I

I flip two fair coins. Let X count the number of heads, and let Y be an indicator for the first coin being a head. First, what is XY ?

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Example: Coin Flips I

What is Cov(X, Y )?

9 / 27

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Properties of Covariance I

If X and Y are independent, then: Cov(X, Y ) = Is the converse true?

10 / 27

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Properties of Covariance II

What happens when we take Cov(X, X)? Can use either definition of covariance!

11 / 27 if ECX]

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Properties of Covariance III

Covariance is bilinear. Cov(a1X1 + a2X2, Y ) = Cov(X, b1Y1 + b2Y2) =

12 / 27 A , CoV C Xi , Y ) t Az for ( Xz , Y ) b.for I X , Yi )

tbz

COV ( X , Yz )

④

var C CX ) = Cov ( CX , CX ) = c Cove x. ex ) = Cz Cov C X , X ) .
slide-3
SLIDE 3

Practice: Bilinearity!

Simplify Cov(3X + 4Y , 5X − 2Y ).

13 / 27

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Var l Y)

Example: Coin Flips II

Same setup: Two fair flips. X is number of heads, and Y is indicator for the first coin heads. Recall: Cov(X, Y ) = Now, let Y 0 be an indicator for the first coin being a tail. How does the covariance change?

14 / 27 1-

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Properties of Covariance IV

For any two RVs X, Y : Var(X + Y ) =

15 / 27 Cov ( Xt Y , Xty ) = Cov ( X , Xty ) + COVEY , Xt Y) = Cov ( X , x ) t Cov C Xi ) t COV CY , X ) TCOVCY ,'D = vortex,

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Break

If you could eliminate one food so that no one would eat it ever again, what would you pick to destroy?

16 / 27

Correlation

For any two RVs X, Y that are not constant: Corr(X, Y ) = Sanity Check! What is Corr(X, X)? What is Corr(X, −X)? What is Corr(X, Y ) for X, Y independent?

17 / 27 so

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Example: Coin Flips II

I flip two fair coins. Let X count the number of heads, and let Y be an indicator for the first coin being a head. What is Corr(X, Y )?

18 / 27 µ ↳ =

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slide-4
SLIDE 4

Size of Correlation?

For any RVs X and Y that are not constants: −1 ≤ Corr(X, Y ) ≤ 1 Proof: Define new RVs using X and Y : ˜ X = ˜ Y =

19 / 27

Size of Correlation?

(Continued:)

20 / 27

RV Practice: Two Roads

There are two paths from Soda to VLSB. I usually choose a path uniformly at random. # minutes spent on Path 1 is a Geometric(p1) RV. # minutes spent on Path 2 is a Geometric(p2) RV. Today, it took me 6 minutes to walk from Soda to VLSB. Given this, what is the probability that I chose Path 1?

21 / 27
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RV Practice: Two Roads

Continued:

22 / 27

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RV Practice: RandomSort

I have cards labeled 1, 2, . . . , n. They are shuffled. I want them in order. I sort them in a naive way. I start with all cards in an “unsorted” pile. I draw cards from the unsorted pile uniformly at random until I get card 1. I place card 1 in a “sorted” pile, and continue, this time looking for card 2.

23 / 27 goaertd I 9

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re

a What are these deists ?

RV Practice: RandomSort

What is the expected number of draws I need? What is the variance of the number of draws?

24 / 27
slide-5
SLIDE 5

RV Practice: Packets

Packets arrive from sources A and B. I fix a time interval. Over this interval, the number

  • f packets from A and B have Poisson(λA) and

Poisson(λB) distributions, and are independent. What is the distribution of the total number of packets I receive in this time interval?

25 / 27

RV Practice: Packets

What is the probability that over this interval, I receive exactly 2 packets? What is the expected number of packets I receive

  • ver this interval?

What is the variance of this number?

26 / 27

Summary

I Covariance and correlation measure how independent two RVs are. I Variance can be expressed and manipulated in terms of covariance. I Independent RVs have zero covariance and zero correlation. However, the converse is not true!

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