Property of expectation

Question

Give X is a random variable , u(X) is a function of X and c is a constant,

is it true that $E(c[u(X)]) = c[E(u(X))]$

If is it true, can someone provide me with a short proof?

Thanks in advance!

Answer 1

For the discrete case:$$E(c[u(X)])=\sum_{i=1}^ncu(x_i) \mathbb P(X=x_i)=c\sum_{i=1}^nu(x_i) \mathbb P(X=x_i)=cE([u(X)]).$$ The proof for the continuous case is the same but with an integral.