Solutions - 5250-05

(a)

$$\begin{array}{cccc}x& 0& 1& 2\\ P_X(x)& {0.3}^2& 2(0.7)(0.3)& {0.7}^2\\ & =0.09& =0.42& =0.49\end{array}$$

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(b)

$$\begin{array}{cc}y& 0\\ P_{Y|X}(y|0)& 1\end{array}\begin{array}{ccc}y& 0& 1\\ P_{Y|X}(y|1)& 0.85& 0.15\end{array}\begin{array}{cccc}y& 0& 1& 2\\ P_{Y|X}(y|2)& {(0.85)}^2& 2(0.85)(0.15)& {0.15}^2\end{array}$$

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(c)

Multiply $P_{Y|X}(y|x)\cdot P_X(x)$ for each cell:

$$\begin{array}{cccc}Y\downarrow X\to & 0& 1& 2\\ 0& 0.09& \begin{array}{c}0.42(0.85)\\ =0.357\end{array}& \begin{array}{c}0.49({0.85}^2)\\ =0.354\end{array}\\ 1& 0& \begin{array}{c}0.42(0.15)\\ =0.063\end{array}& \begin{array}{c}0.49\cdot 2(0.85)(0.15)\\ =0.125\end{array}\\ 2& 0& 0& \begin{array}{c}0.49{(0.15)}^2=0.011\end{array}\end{array}$$