Solutions - 5190-12

(a)

(1) Tabulate the joint PMF for small values of $Y$:

$$\begin{array}{cccc}X\downarrow Y\to & 1& 2& 3\\ 1& \frac{.05}{1}{(.95)}^0& \frac{.05}{2}{(.95)}^1& \frac{.05}{3}{(.95)}^2\\ 2& 0& \frac{.05}{2}{(.95)}^1& \frac{.05}{3}{(.95)}^2\\ 3& 0& 0& \frac{.05}{3}{(.95)}^2\\ 4& 0& 0& 0\end{array}$$

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(2) Sum over $Y=1$ and $Y=2$:

$$\begin{array}{c}P[Y<3]=P[Y=1]+P[Y=2]\\ \\ \gg \gg \frac{.05}{1}{(.95)}^0+2\left(\frac{.05}{2}{(.95)}^1\right)\gg \gg .0975\end{array}$$

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(b)

$$\begin{array}{c}P[X=2]=P[X=1]-P_{X,Y}(\mathrm{1,1})\\ \\ \gg \gg .1577-.05\gg \gg .1077\end{array}$$