Expectation of a R.V times a Constant C.

Question

supposed that we have to deal with this expectation:

let X be a R.V. and C some constant.

can we say that:

$E[CX]= E[C]E[X] = CE[X]$

Thanks!

Answer 1

You can say that in the understanding that $C$ following the symbol $\mathsf E$ denotes a random variable prescribed by $\omega\mapsto C$ and is defined on the same probability space as $X$.

In that case $X$ and $C$ are automatically independent random variables and $\mathsf EC=C$ where the LHS is the expectation of a random variable and the RHS is a constant.

That leads to:$$\mathsf ECX=\mathsf EC\mathsf EX=C\mathsf EX$$

Answer 2

Note that $$ EcX=\int cX(w)\, dP(w)=c\int X(w)\,dP(w)=cEX $$ and $$ Ec=\int c\,dP=c\int\,dP=c. $$