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amentals of Probability and Statistics \( > \) Files \( > \) test-1-practise.pdf
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Problem \( \Lambda \) health-food store stocks two different brands of grain. Let \( Y_{1} \) denote the amount of brand 1 in stock and let \( Y_{2} \) denote the amount of brand 2 in stock (both \( Y_{1} \) and \( Y_{2} \) are measured in 100s of ths). The joint distribution of \( Y_{1} \) and \( Y_{2} \) is given by
\[
f(y)=\left\{\begin{array}{ll}
20 y_{1} y_{2} & y_{1}>0, y_{2}>0,0<y_{1}+y_{2}<1 \\
0 & \text { otherwise }
\end{array}\right.
\]
(a) Find the conditional pif \( f_{Y_{2} \mid y_{1}}\left(y_{2} \mid y_{1}\right) \).
(b) Find \( P\left(Y_{2}>0.5 \mid Y_{1}=0.2\right) \).
(c) Find \( P\left(Y_{2}>0.5\right) \).
(d) Find \( P\left(Y_{1}<0.7, Y_{2}>0.5\right) \).
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