Q:

How to prove that $\mathbb{E}(c*X)=c*\mathbb{E}(X)$ using the definition of expectation $\mathbb{E}(X)$

Please someone help me to prove the following theorem:
$$\mathbb{E}(c*X)=c*\mathbb{E}(X)$$
using the definition of expectation $$\mathbb{E}(X)$$:
$$\mathbb{E}(X) = \sum_{i=1}^{n}x_iP(x_i)$$

A:

Easy. Let $X$ be a random variable with PMF $f(x)$ on the sample space $\Omega$. Now let $c$ be a scalar. Then
\begin{align*}
E(cX) &= \sum_{x \in \Omega} cxf(x) \\
&= c\sum_{x \in \Omega} xf(x) \\
&= cE(X)
\end{align*}
Done. Sums and integrals are linear so expectation is linear.

