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

Question

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

Answer 1

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.