I have a problem that seems to be straight forward, but I can't seem to construct the proof when I think through.
Problem:
Let X be a random variable supported on [0,1], and Ber($p$) be a Bernoulli random variable with success probability $p$. Is it true that Ber(X) and Ber$(\mathbb{E}[X])$ have the same distribution?
Knowing that $\mathbb{E}[X]$ for Bernoulli is $p(1)+ (1-p)(0) = p$, a constant, it is obvious that the two will have the same distribution, but how do I show this through a proof?