Problems

01

Conditional distribution and expectation from joint PMF

Suppose that $X$ and $Y$ have the following joint PMF:

$$P_{X,Y}(k,\ell )=\{\begin{array}{cc}c& k=\mathrm{1,2,3,4};\ell =1,\dots ,k\\ 0& \text{ otherwise }\end{array}$$

Notice that the possibilities for $\ell$ depend on the choice of $k$.

First, show that $c=1/10$. Then compute:

(a) $P_X$ $$ (b) $P_{Y|X}$ $$ (c) $E[Y|X=4]$ $$ (d) $E[Y|X]$

02

Conditional distribution and expectation from joint PDF

Suppose that $X$ and $Y$ have the following joint PDF:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}cxy& 0<y<1,0<x<y\\ 0& \text{ otherwise }\end{array}$$

Notice that the range of possibilities for $x$ depends on the value of $y$.

First, show that $c=8$. Then compute:

(a) $f_X$ $$ (b) $f_{Y|X}$ $$ (c) $E[Y|X=0.5]$ $$ (d) $E[Y|X]$

03

“Plug In” Expectation Identity

Suppose $h(x,y)$ is a function, and $X$ and $Y$ are two random variables.

Verify this formula in the continuous case, using the definitions:

$$E[h(X,Y)|Y=y]=E[h(X,y)|Y=y]$$

Using that formula, prove this formula:

$$E[a(Y)b(X)|Y]=a(Y)E[b(X)|Y]$$

for two functions $a(y)$ and $b(x)$ and random variables $X$ and $Y$. Notice that here the expectations are viewed as random variables.

Hint for second question: Both sides are functions of $Y$. Write these functions as $g_1(Y)$ and $g_2(Y)$ and check equality of the functions.

04

Expectation Multiplication Rule

Prove the following identity using Iterated Expectation along with the previous exercise:

$$E[XY]=E[YE[X|Y]]$$

Note: The solution is short once you find it – please clearly identify your choices for $a(y)$ and $b(x)$ functions.

05

How many customers buy a cake?

Let $N$ count the number of customers that visit a bakery on a random day, and assume $N\sim \mathrm{Pois}(\lambda )$.

Let $X$ count the number of customers that make a purchase. Each customer entering the bakery smells the cakes, and this produces a probability $p$ of buying a cake for that customer. The customers are independent.

Find $\mathrm{Cov}[N,X]$. Are $N$ and $X$ positively or negatively correlated?

Hint: Compute $P_{X|N}(x|n)$, and use this to compute $E[X|N]$ in terms of $N$. Now deduce $E[X]$ using Iterated Expectation. Finally, compute $E[NX]$ using the Expectation Multiplication Rule from the previous exercise. Now put everything together to find $\mathrm{Cov}[N,X]$.