I am self-learning basic undergrad calculus-based probability. I would like someone to verify if my solution to the below exercise problem on the total expected value is correct.
[BH 4.49] There are $n$ prizes, with values \$1, \$2, $\ldots$, \$n. You get to choose $k$ random prizes without replacement. What is the expected total value of the prizes you get?
Hint: Express the total value in the form $a_1 I_1 + a_2 I_2 + \ldots + a_n I_n$, where the $a_j$ are constants and the $I_j$ are indicator random variables. Or find the expected value of the $j$th prize received directly.
Solution. (My Attempt)
Let $A_j$ be the event that the $j$th prize is included in the sample of size $k$ and let $I_{A_j}$ be its indicator function.
\begin{align*} P(A_j) = \frac{n-1 \choose k - 1}{n \choose k} = \frac{k}{n} \end{align*}
Let $X$ be the total value of the prize. Then,
\begin{align*} X &= 1\cdot I_{A_1} + 2 \cdot I_{A_2} + \ldots + n\cdot I_{A_n}\\ E(X) &= \frac{k}{n}\sum_{j=1}^{n}j\\ &= \frac{k}{n}\cdot \frac{n(n+1)}{2}\\ &= \frac{k(n+1)}{2} \end{align*}
The expected value of the total prize checks out for the edge cases $k=1$ and $k=n$.