Properties of Mathematical expectation and variance

Three linearity/scaling laws hold for any random variable $X$ and constants $a,b$.

(i) $E(aX+b)=aE(X)+b$.

Proof (discrete case; the continuous case is identical with integrals in place of sums). $E(aX+b)=\sum_i(ax_i+b)f(x_i)=a\sum_i x_if(x_i)+b\sum_i f(x_i)=aE(X)+b\cdot1=aE(X)+b$, using $\sum_i f(x_i)=1$.

• Corollary 1 ($b=0$): $E(aX)=aE(X)$.
• Corollary 2 ($a=0$): $E(b)=b$ — the expectation of a constant is the constant itself.

(ii) $V(X)=E(X^2)-\big(E(X)\big)^2$.

Proof. With $\mu=E(X)$: $V(X)=E((X-\mu)^2)=E(X^2-2\mu X+\mu^2)=E(X^2)-2\mu E(X)+\mu^2=E(X^2)-2\mu^2+\mu^2=E(X^2)-\mu^2$ (using $\mu$ constant and part (i)).

(iii) $V(aX+b)=a^2V(X)$.

Proof. $V(aX+b)=E\big((aX+b-E(aX+b))^2\big)=E\big((aX+b-aE(X)-b)^2\big)=E\big(a^2(X-E(X))^2\big)=a^2E\big((X-E(X))^2\big)=a^2V(X)$. …