Let $X$ be some random variable. Now define a new random variable $Y$ such that it has a probability $p$ of taking some fixed number $a$, and a probability $(1-p)$ of being determined by $X$.
What is the expected value of $Y$? Is it true that it is just $E[Y] = pa + (1-p)E[X]$? If this is the case, how do I prove this?