What are the density functions for the random variables $2X+1$ and $ X^2$ knowing the density function for variable $X$.

Question

This is the density function for random variable $X$: $$f(x)=\begin{cases}2x&\text{ if }& 0\leq x\leq 1\\0&\text{ otherwise }.\end{cases}$$

Compute the density functions for $2X+1$ and $X^2$.

Answer 1

Both functions are bijective over the support, so it is a straight forward application of the Jacobian transformation theorem.

$$f_{g(X)}(y) = \begin{vmatrix}\dfrac{\partial~g^{\small-1}(y)}{\partial~y\hspace{5ex}}\end{vmatrix}\;f_X(g^{\small-1}(y))$$

Where you have

• $g(x)=(2x+1)~\iff~ g^{\small-1}(y)=(y-1)/2$
• $g(x)=x^2 \hspace{6.5ex}\iff~ g^{\small-1}(y)=\surd y$