W10 Homework B

Due date: Sunday 3/22, 11:59pm

Covariance and correlation

01 $★$

04

Correlation between overlapping coin flip sequences

Suppose a coin is flipped 30 times.

Let $X$ count the number of heads among the first 20 flips, and $Y$ count the heads in the last 20.

Find $\rho [X,Y]$.

Hint: Partition the flips into three groups of 10. Create three variables, counting heads, and express $X$ and $Y$ using these. What is the variance of a binomial distribution?

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Solution

Solutions---5240-04

02 $★$

05

Variance puzzle: indicators

Suppose $A$ and $B$ are events satisfying:

$$P[A]=0.5,P[B]=0.2,P[AB]=0.1$$

Let $X$ count the number of these events that occur. (So the possible values are $X=\mathrm{0,1,2}$.)

Find $\mathrm{Var}[X]$.

Hint: Try setting $X=X_A+X_B$.

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Solution

Solutions---5240-05

03

07

Covariance etc. from joint density

Suppose $X$ and $Y$ are random variables with the following joint density:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}\frac{3}{2}(x^2+y^2)& x,y\in [\mathrm{0,1}]\\ 0& \text{ otherwise }\end{array}$$

Compute:

(a) $E[X]$ $$ (b) $E[Y]$ $$ (c) $E[XY]$ $$ (d) $\mathrm{Var}[X]$

(e) $\mathrm{Var}[Y]$ $$ (f) $\mathrm{Cov}[X,Y]$ $$ (g) $\rho [X,Y]$ $$ (h) Are $X$ and $Y$ independent?

(It is worth thinking through which of these can be computed in multiple ways.)

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Solution

Solutions---5240-07

Optional challenge problem

The next problem is purely optional, but worthwhile for the stouthearted. The result is interesting and meaningful.

04

06

When $\rho [X,Y]=1$

Suppose $\rho [X,Y]=1$ for two random variables $X$ and $Y$.

Prove that $Y=aX+b$, where $a={\sigma }_Y/{\sigma }_X$, and find the formula for $b$.

Hint: Study the derivation that $-1\le \rho [X]\le 1$, and think about $E[{(\tilde{X}-\tilde{Y})}^2]$.

(Note: A similar result and argument holds for $\rho [X,Y]=-1$.)

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Solution

Solutions---5240-06

Review

Independent random variables

05

02

Factorizing the density

Consider two joint density functions for $X$ and $Y$:

$$\begin{array}{cc}{f}_1(x,y)& =6e^{-2x}{e}^{-3y}x,y>0,\\ f_2(x,y)& =24xyx,y\in [\mathrm{0,1}],x+y\in [\mathrm{0,1}].\end{array}$$

(Assume the densities are zero outside the given domain.)

Supposing $f_1$ is the joint density, are $X$ and $Y$ independent? Why or why not?

Supposing $f_2$ is the joint density, are $X$ and $Y$ independent? Why or why not?

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Solution

Solutions---5200-02