Normal variables - adding and multiplying by constant

Question

$X\sim N(a,b)$, while $c$ is constant.

Is is true that then:

1. $X+c\sim N(a+c,b)$ ?
2. $cX\sim N(c\cdot a, b)$ ?

Answer 1

Let $X\sim \mathcal{N}(a,b)$. Let $c > 0$.

Then, $X+c \sim \mathcal{N}(a+c,b)$ and $cX \sim \mathcal{N}(ca,c^2 b)$.

Answer 2

The first statement is true. The second statement is false.

We have that $$ F_X(x)=\int_{-\infty}^x\frac{1}{\sqrt{2b\pi} } \; e^{ -\frac{(t-a)^2}{2b} }\mathrm dt $$ and \begin{align*} F_{X+c}(x) &=P(X+c\le x)\\ &=P(X\le x-c)\\ &=\int_{-\infty}^{x-c}\frac{1}{\sqrt{2b\pi} } \; e^{ -\frac{(t-a)^2}{2b} }\mathrm dt\\ &=\int_{-\infty}^x\frac{1}{\sqrt{2b\pi} } \; e^{ -\frac{(s-c-a)^2}{2b} }\mathrm d(s-c)\\ &=\int_{-\infty}^x\frac{1}{\sqrt{2b\pi} } \; e^{ -\frac{(s-(a+c))^2}{2b} }\mathrm ds. \end{align*} Hence, $X+c\sim\mathcal N(a+c,b)$.

To see that the second statement is false, calculate the variance $\operatorname{Var}[cX]$.