The effect of scaling arbitrary random varaible on its probability distribution?

Question

Is it true that for an arbitrary random variable $X$, the distribution of $\frac{X}{2}$ is simply horizontally squeezing the original distribution to half of the original size and stretching it vertically to double of the original size? I mean strictly speaking rather than just a hand-waving ilustration.

Answer 1

Let $X$ be a continuous, real valued random variable with probability density function (PDF) $f_X$, and denote $Y:=X/2$.

We will compute the PDF of $Y$ $f_Y$, which is given by the derivative of its CDF $F_Y$. For all $t\in\mathbb R$ we have : $$F_Y(t) := \mathbb P(Y\le t) = \mathbb P(X/2\le t) = \mathbb P(X\le 2t) = F_X(2t) \tag1$$

Where $F_X$ is the CDF of $X$. It thus follows by differentiating $(1)$ w.r.t. $t$ and applying the chain rule that for all $t\in\mathbb R$ : $$f_Y(t) =\frac{d}{dt}F_Y(t) = \frac{d}{dt}F_X(2t) = \color{red}{2}f_X(\color{blue}{2}t) $$

We thus see that, indeed, the PDF of $X/2$ is that of $X$ $\color{red}{\text{stretched vertically}} $ by a factor of $2$ and $\color{blue}{\text{squeezed horizontally}} $ by a factor of $2$.