I am self-learning basic undergrad calculus-based probability. I would like someone to verify if my solution is correct, and I didn't make any unreasonable assumptions. I am posting my attempt/solution below:
Let $X \sim Bin(n,\frac{1}{2})$ and $Y \sim (n+1,\frac{1}{2})$ independently.
(a) Let $V = \min(X,Y)$ be the smaller of $X,Y$ and let $W=\max(X,Y)$ be the larger of $X,Y$. So, if $X$ crystallizes to $x$ and $Y$ crystallizes to $y$, then $V$ crystallizes to $\min(x,y)$ and $W$ crystallizes to $\max(x,y)$. Find $E(V) + E(W)$.
Solution. (My Attempt)
The PMF of the random variables $V,W$ can be deduced, but it can be computationally very laborious.
Observe that, the sum of the rvs, $V + W = X + Y$, the total number of successes in both the experiments.
So,
\begin{align*} E(V) + E(W) &= E(V + W)\\ &= E(X + Y) \\ &= E(X) + E(Y)\\ &= \frac{n}{2} + \frac{n+1}{2}\\ &= \frac{2n+1}{2} \end{align*}