# Transformations of random variables with the uniform distribution

• Nov 26th 2009, 01:22 PM
chella182
Transformations of random variables with the uniform distribution
This is probably hellishly easy, but I'm just not getting it.

$X$ is uniformly distributed over $(0,1)$. For the random variable $Y=-log_{e}(X)$
i) State the range of values $Y$ can take.
ii) Obtain the distribution function of $Y$, $F_{Y}(y)$.
iii) Obtain and identify the probability density function of $Y$, $f_{Y}(y)$.

Cheers for any help in advance.
• Nov 26th 2009, 02:50 PM
matheagle
The range is $(1,\infty)$

Let y exceed 1...

$F_Y(y)=P(Y\le y)=P(-\ln X\le y)=P(X\ge e^{-y})$

$=\int_{e^{-y}}^1 dx=1-e^{-y}$ which is EXP(1)