Solutions - 5260-01

Find the possible combinations of $k$ and $\ell$. For a given value of $k$, there are $k$ possible options for $\ell$. In total there are $1+2+3+4=10$ combinations, so $c=\frac{1}{10}$.

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(a)

$$P_X(k)=\{\begin{array}{cc}{\displaystyle \frac{k}{10}}& k=1,2,3,4\\ 0& \text{otherwise}\end{array}$$

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(b)

(1) Apply the conditional PMF formula:

$$\begin{array}{c}{P}_{Y|X}(\ell |k)=\frac{P_{X,Y}(k,\ell )}{P_X(k)}\\ \\ \gg \gg \frac{c}{kc}\gg \gg \frac{1}{k}\end{array}$$

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(2) State the conditional PMF:

$$P_{Y|X}(\ell |k)=\{\begin{array}{cc}{\displaystyle \frac{1}{k}}& \ell =1,\dots ,k\\ 0& \text{otherwise}\end{array}$$

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(c)

Compute the conditional expectation:

$$\begin{array}{c}E[Y|X=4]=\sum _{\ell =1}^4\frac{\ell }{4}\gg \gg {\displaystyle \frac{5}{2}}\end{array}$$

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(d)

Find the general formula for conditional expectation:

$$\begin{array}{c}E[Y|X]=\sum _{\ell =1}^x\frac{\ell }{x}\\ \\ \gg \gg \frac{1}{x}\sum _{\ell =1}^x\ell \gg \gg \frac{1}{x}\cdot \frac{x(x+1)}{2}\gg \gg {\displaystyle \frac{x+1}{2}}\end{array}$$