Question

1. Let D be a discrete random variable with the following pmif 1/10 ifk-б 1/10 ifk-б 2/5 if k7 1/10 if k9 otherwise. Find a) E min(D, 7). a) E[(7 - D). (Recall that (7 - D)+ax7 - D, 0).

Answer 1

a)$\begin{align*} E[min(D,7)] &= \sum_{k=5}^9 min(k,7)*P(D=k) \\ &= min(5,7)*P(D=5)+min(6,7)*P(D=6)+min(7,7)*P(D=7) \\ & \ \ \ +min(8,7)*P(D=8)+min(9,7)*P(D=9) \\ &= 5*\frac{1}{10} + 6*\frac{1}{10} + 7*\frac{2}{5} + 7*\frac{3}{10} + 7*\frac{1}{10} \\ &= \frac{5+6+28+21+7}{10} \\ &= \textbf{6.7 \ \ \ \ \ \ \ \ \ [Answer]} \end{align*}$

b)

$\begin{align*} E[(7-D)^+] &= E[max(7-D,0)] \\ &= \sum_{k=5}^9 max(7-k,0)*P(D=k) \\ &= max(7-5,0)*P(D=5)+max(7-6,0)*P(D=6)+max(7-7,0)*P(D=7) \\ & \ \ \ +max(7-8,0)*P(D=8)+max(7-9,0)*P(D=9) \\ &= 2*\frac{1}{10} + 1*\frac{1}{10} + 0*\frac{2}{5} + 0*\frac{3}{10} + 0*\frac{1}{10} \\ &= \frac{3}{10} \\ &= \textbf{0.3 \ \ \ \ \ \ \ \ \ [Answer]} \end{align*}$

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