Solutions - 5190-09

	$Y=1$	$Y=2$	$Y=3$	$P_X(k)$
$X=1$	0.09	0.11	0.05	0.25
$X=2$	0.17	0.23	0.08	0.48
$X=3$	0.06	0.15	0.06	0.27
$P_Y(\ell )$	0.32	0.49	0.19

(a)

Sum the joint probabilities where $X>Y$:

$$\begin{array}{c}P[X=2,Y=1]+P[X=3,Y=1]+P[X=3,Y=2]\\ \\ \gg \gg 0.17+0.06+0.15\gg \gg 0.38\end{array}$$

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(b)

Compute $E[X]$, $E[Y]$, and use linearity:

$$\begin{array}{c}E[X]=0.25+2(0.48)+3(0.27)\gg \gg 2.02\\ \\ E[Y]=0.32+2(0.49)+3(0.19)\gg \gg 1.87\\ \\ E[X+Y]=E[X]+E[Y]\gg \gg 3.89\end{array}$$