Suppose X is a continuous random variable with probability density $\displaystyle f$, and $\displaystyle \phi$ is a continuous increasing differentiable function.

1) Show that the probability density of $\displaystyle g$ of $\displaystyle Y=\phi(X)$ is given by:

$\displaystyle g(y)=f(\phi^{-1}(y))\psi(y)$

where $\displaystyle \phi^{-1}$ is the inverse of $\displaystyle \phi$, and

$\displaystyle \psi(y)=\frac{d}{dy}\phi^{-1}(y)$

2) Show that the expected value of Y is:

$\displaystyle E(Y)=\int_{-\infty}^{\infty} \phi(x)f(x) dx$