Normal Distribution PDF conversion

wolverine21 · #1 $Old$ November 18th, 2008, 12:17 AM

A) Find the pdf of W = X^2 when X ~ N(10,9)

B) Suppose that X ~ N(E(X)), Var(X)). Find the pdf of Y = e^x

Not sure how to do these. Any help?

mr fantastic · #2 $Old$ November 18th, 2008, 03:28 AM

Quote:

Originally Posted by wolverine21 $View Post$

A) Find the pdf of W = X^2 when X ~ N(10,9)

B) Suppose that X ~ N(E(X)), Var(X)). Find the pdf of Y = e^x

Not sure how to do these. Any help?

A) Similar to the one asked here: Density Functions

B) Get the cdf of Y: $G(y) = \Pr(Y < y) = \Pr(e^X < y) = \Pr(X < \ln y) = \, ....$

Then the pdf of Y is $g(y) = \frac{dG}{dy}$ .

Note: $G(y) = \int_{-\infty}^{\ln y} f(x) \, dx \Rightarrow \frac{dG}{dy} = \frac{1}{y} \, f(\ln y)$ (using the Fundamental Theorem of Calculus and the chain rule) where $f(x)$ is the pdf of X.