I have 2 related questions that I am stuck on, in particular, there is the following:
"Let X be a random variable with density function $f_{X}$ , and let $Y
> = X + b$, $Z = aX$, and $W = aX + b$, where $a \neq 0$. Find the density functions $f_{Y}$ , $f_{Z}$, and $f_{W}$"As a follow up, the 2nd goes as follows
"Let X be a random variable with cumulative distribution function $F_{X}$, and let $Y=X+b$,$Z=aX$,and $W =aX+b$,where $a$ and $b$ are any constants. Find the cumulative distribution functions $F_{Y}$ , $F_{Z}$ , and $F_{W}$"
I would be able to figure out the two if one just shows me how to do the case for $a > 0$ because I think it is quite analogous. However, I am quite stuck on these two questions. I know that for the former question, the answer is given by
$a \neq 0: f_{W}(w) = \frac{1}{|a|}*f_{X}(\frac{w-b}{a})$ but I am not quite sure how this result is derived and would like some help. Would greatly appreciate any sort of help.
Thank you.
Best Answer
In general if $U=g(X)$ where $g$ is a bijection mapping the support of $X$ to the support of $U$ we apply a change of variables transformation:
$$ f_U(u) = f_X(g^{-1}(u))\cdot\left\lvert\frac{\mathrm d g^{-1}(u)}{\mathrm d u}\right\rvert$$
[Note: the modulus is because density is an unsigned measure of the gradient of the CDF.]
But for these three, it is just a linear scale and shift transformation:
$Y$ is $X$ shifted by $b$. So the density of $Y$ at $y$ is that of $X$ at $x-b$.
$$f_Y(y) = f_X(y-b)$$
$Z$ is $X$ scaled by $a$. So the density of $Z$ at $z$ is $\lvert 1/a\rvert$ times the density of $X$ at $z/a$. $$f_Z(z) = \tfrac 1 {\lvert a\rvert}~f_X(\tfrac z a)$$
$W$ is $X$ scaled by $a$ and shifted by $b$. So...
$$f_W(w) = \tfrac 1 {\lvert a\rvert}\cdot f_X\left(\tfrac{w-b}{a}\right)$$
NB: as stated, the density function must be non-negative.
$$F_Y(y) = \Pr(X+b\leq y) = F_X(y-b)$$
$$F_Z(z) = \Pr(aX\leq z) = \begin{cases}F_X(z/a) & : a>0 \\ 1-F_X(z/a) & : a< 0 \\ \mathbf 1_{0\leq z} & : a=0\end{cases}$$
$$F_W(w) = ....$$