# Find the pdf

• Aug 26th 2010, 09:20 PM
jlt1209
Find the pdf
Let X be a Uniform(0,1) random variable. Find the pdf of Y = (-1/lamda)*ln(X) , where lamda is a given positive number.

I don't think this is very difficult, but I'm not really sure how to go about it when X and Y are together like that. If anyone could point me in the right direction I'd really appreciate it!
• Aug 26th 2010, 10:36 PM
aman_cc
Hint -
1. Try plotting y as f(x)? What is the range it takes?
2. Can you find the cdf for y? i.e. Prob (y<y0)
3. Differentiate the cdf to get pdf

This should not be too tough
• Aug 26th 2010, 10:49 PM
matheagle
Use Calc 1

$f_Y(y)=f_X(x)\left|{dx\over dy}\right|$
• Aug 27th 2010, 03:49 AM
mr fantastic
Quote:

Originally Posted by jlt1209
Let X be a Uniform(0,1) random variable. Find the pdf of Y = (-1/lamda)*ln(X) , where lamda is a given positive number.

I don't think this is very difficult, but I'm not really sure how to go about it when X and Y are together like that. If anyone could point me in the right direction I'd really appreciate it!

Quote:

Originally Posted by matheagle
Use Calc 1

$f_Y(y)=f_X(x)\left|{dx\over dy}\right|$

Alternatively, since X ~ U(0, 1) => 1 - X ~ U(0, 1) it follows immediately from the probability integral transform theorem that Y is an exponential random variable with pdf $\lambda e^{- \lambda y}$.