1. ## Exponential Random Variables

Let X be an exponential random variable with mean 1. Find the probability density function of $Y = -ln (X)$.

Can someone walk me through how to do this?

Exponential random variable has a density function $f(x) = \lambda e^{-\lambda x}$. In this case $\lambda = 1$. How do I find that density function of Y?

2. Originally Posted by Zennie
Let X be an exponential random variable with mean 1. Find the probability density function of $Y = -ln (X)$.

Can someone walk me through how to do this?

Exponential random variable has a density function $f(x) = \lambda e^{-\lambda x}$. In this case $\lambda = 1$. How do I find that density function of Y?
What techniques have you been taught for finding the pdf of a function of a random variable? Do you have any examples to follow? What have you tried and where do you get stuck?

3. Originally Posted by Zennie
Let X be an exponential random variable with mean 1. Find the probability density function of $Y = -ln (X)$.

Can someone walk me through how to do this?

Exponential random variable has a density function $f(x) = \lambda e^{-\lambda x}$. In this case $\lambda = 1$. How do I find that density function of Y?
Hint: the cdf of Y is $F_{Y}(y)= P(Y\leq y)$

now use $Y=-\ln(X)$ above to find the cdf in terms of x. Then differentiate to find the pdf!

4. Originally Posted by Zennie
Let X be an exponential random variable with mean 1. Find the probability density function of $Y = -ln (X)$.

Can someone walk me through how to do this?

Exponential random variable has a density function $f(x) = \lambda e^{-\lambda x}$. In this case $\lambda = 1$. How do I find that density function of Y?
Assume that $R=f(x)$ in this case $R = \lambda e^{-\lambda x}$
we need to proof that $x=F(R).$
let's do it:
$
1-R= \lambda e^{-\lambda x}$

$Ln(1-R)=Ln( \lambda e^{-\lambda x})$
$Ln(1-R)=-\lambda x Ln( e) ..........Ln(e)=1$
x= (-1/ lambda) x Ln(1-R)......... Ln(1-R) same as Ln(R) CZ R<1 , R IS UNIFROM DISTRIBUTION OVER [0,1]
x= (-1/ lambda) x Ln(R)
.... THIS IS INVERSE TRANSFORM THECHNIQUE
HOPE THAT HELPS
ABDEL