Requirements for random variables to be distributed identically

Question

If two random variables have the same expected value and the same variance, are they distributed identically? If not, how can we test if two random variables are identically distrubuted?

Many thanks in advance

Rolandos

Answer 1

No. If the 2 random variables with equal mean and variance may not have equal PDF. Consider for example:$$X\sim N(\mu,\sigma^2)$$and$$Y\sim U(\mu -\sigma\sqrt 3,\mu +\sigma\sqrt 3)$$where $\sigma>0$ is the mean deviation.

One standard way way to check whether two random variables are identically distributed if their Moment Generating Functions are equal, i.e.:$$E\{e^{Xt}\}=E\{e^{Yt}\}$$