Convexity of expected value

Question

I am trying to understand if the expected value of a variable is convex in that variable or not. I know that expectation is a linear operator, so must be convex. But I do not see why it does not depend on what characteristics the probability vector $p(x)$ has. In other words, why $E(x)= x^Tp(x)$ is convex for any $p(.)$ such that $p\ge0, p(x)^T1=1$? Is $p$ constant wrt $x$? Confused.

Answer 1

The expected value does not depend on $x$ as we integrate out $x$. So, your question is not meaningful.