A 1-rupee coin,a 2-rupee coin,a 5-rupee coin, | vedclass

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(D) Let $X_1, X_2, X_3, X_4$ be the values of the coins $1, 2, 5, 10$ respectively. Let $I_k$ be an indicator random variable such that $I_k = 1$ if t

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$9$

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(D) Let $X_1, X_2, X_3, X_4$ be the values of the coins $1, 2, 5, 10$ respectively. Let $I_k$ be an indicator random variable such that $I_k = 1$ if the $k$-th coin shows heads and $I_k = 0$ if it shows tails. The probability of getting heads for each coin is $P(I_k = 1) = \frac{1}{2}$. The expected value of each indicator variable is $E[I_k] = 1 	imes \frac{1}{2} + 0 	imes \frac{1}{2} = \frac{1}{2}$. The total sum $S$ of the values of the coins showing heads is $S = 1 \cdot I_1 + 2 \cdot I_2 + 5 \cdot I_3 + 10 \cdot I_4$. By the linearity of expectation,$E[S] = 1 \cdot E[I_1] + 2 \cdot E[I_2] + 5 \cdot E[I_3] + 10 \cdot E[I_4]$. $E[S] = 1 	imes \frac{1}{2} + 2 	imes \frac{1}{2} + 5 	imes \frac{1}{2} + 10 	imes \frac{1}{2}$. $E[S] = \frac{1+2+5+10}{2} = \frac{18}{2} = 9$.

$X$	$1$	$2$	$3$	$4$	$5$
$P(X = x)$	$k$	$\frac{k}{3}$	$\frac{k}{4}$	$\frac{k}{2}$	$\frac{k}{2}$

$X=x$	$-1$	$0$	$1$	$2$
$P(X=x)$	$\frac{1}{3}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{3}$

$X = x_i$	$1$	$2$	$3$	$5$
$P(X = x_i)$	$2k^2$	$k$	$k$	$k^2$

$A$ $1$-rupee coin,a $2$-rupee coin,a $5$-rupee coin,and a $10$-rupee coin are tossed simultaneously. The expected value of the sum of the values of the coins that show heads up is:

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