Variance and expected value of maximum?

Question

How to compute the expected value and variance of

\begin{equation} Max(N-B,B) \end{equation}

where $N$ is a constant and $B$ is a binomial variable

$$B\sim B(N,\lambda)$$

also interested in bounds !

Answer 1

Hint: left $A= \max(N-B,B)$, define $X=1$ if $B\ge N/2$, $X=0$ otherwise.

Compute $E[A|X]$ and apply $E[A] = E[E[A|X]]$

Edited:

We have $$ A = \begin{cases} B & \text{if } X=1\\ N-B & \text{if } X=0 \end{cases} $$

Hence we can write $A= X B + (1-X) (N-B)$ and $E[A | X ] = E[B] X + (1-X)(N-E[B])= \lambda X + (1-X) (N-\lambda)$

Then, letting $\mu = E[X] = P(B \ge N/2)$ we have

$$E[A] = E[E[A|X]] = \mu \lambda +(1- \mu)(N-\lambda)$$

To compute $\mu$ you need to evaluate the tail of a Binomial, which has no simple form.