Finding expected value of sample median

Question

Let $X_1,X_2,X_3$ be a random sample from the standard normal distribution. How do I find the expected value of the sample median?

Answer 1

It is not standard normal at all. The probability density function of sample median for odd sample size is given here: look at 4th raw of the table.

For $n=3$ and $k=2$ p.d.f. of second order statistics $X_{(2)}$ equals to $$ f_{X_{(2)}}(x)=6 f(x)\Phi(x)(1-\Phi(x))=6f(x)\Phi(x)\Phi(-x), $$ where $f(x)$ is the p.d.f. of standard normal distribution and $\Phi(x)$ is the cumulative distribution function of standard normal distribution, $$ f(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}=f(-x), $$ and the factor $\Phi(x)\Phi(-x)$ is an even function too.

Then the distribution of $X_{(2)}$ is symmetric, but this is not a standard normal distribution. The expectation exists and equals to zero, as provided in comments.