Answer to Question #275075 in Statistics and Probability for Gotham

Question

Q1. Find the density function of Y = e^x if X is normally distributed with mean μ and standard

deviation σ.

Q2. Find the density function of Y = βX^(1/α) where α > 0, β > 0 if X is exponentially distributed

with mean μ = 1.

Answer 1

1.

density for Normal distribution:

$f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$

 cumulative distribution function:

$F_Y(y)=\int^{lny}_{-\infin}f(x)dx$

To find the density, differentiate. Using the Fundamental Theorem of Calculus:

$f_Y(y)=\frac{1}{y}f(lny)$

$f_Y(y)=\frac{1}{y}\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{lny-\mu}{\sigma})^2}$

2.

$f(x)=\lambda e^{-\lambda x}=e^{-x}$

$X=(Y/\beta)^{\alpha}$

$f_Y(y)=f_X((Y/\beta)^{\alpha})\frac{d}{dy}((Y/\beta)^{\alpha})=\alpha e^{-(y/\beta)^{\alpha}}y^{\alpha -1}/\beta^{\alpha}$