moment generating function of exponential dist.

**lllll** · October 24th 2008, 06:30 PM

I'm having trouble finding the moment generating function of a exponential distribution.

$\phi = \int e^{ty}f(x) dx = \int_{0}^{\infty} e^{ty}\frac{1}{\beta} e^{-y/\beta} dy$

$\phi = \frac{1}{\beta} \int_{0}^{\infty}e^{y(t-1/\beta)} dy$

$\phi = \frac{1}{\beta}\times \frac{1}{t-\frac{1}{\beta}} e^{y(t-1/\beta)}\bigg{|}^{\infty}_{0}$

$\phi= \left(\frac{1}{\beta} \right) \left( \frac{1}{t-\frac{1}{\beta}} \right)$

$\phi = \frac{1}{\beta t-1}$

but the table in the back of my book gives:

$\phi = \frac{1}{\color{red}{1-\beta t}}$

**Chris L T521** · October 24th 2008, 06:43 PM

Originally Posted by lllll

I'm having trouble finding the moment generating function of a exponential distribution.

$\phi = \int e^{ty}f(x) dx = \int_{0}^{\infty} e^{ty}\frac{1}{\beta} e^{-y/\beta} dy$

$\phi = \frac{1}{\beta} \int_{0}^{\infty}e^{y(t-1/\beta)} dy$

$\phi = \frac{1}{\beta}\times \frac{1}{t-\frac{1}{\beta}} e^{y(t-1/\beta)}\bigg{|}^{\infty}_{0}$

$\phi= \left(\frac{1}{\beta} \right) \left( \frac{1}{t-\frac{1}{\beta}} \right)$

$\phi = \frac{1}{\beta t-1}$

but the table in the back of my book gives:

$\phi = \frac{1}{\color{red}{1-\beta t}}$

The issue here in integrating this is the the value of $e^{y(t-1/\beta)}$ as $y\to\infty$ .

We can fix this by writing $e^{y(t-1/\beta)}=e^{-y(1/\beta-t)}$

In evaluating the integral for the moment generating function, we should have

$\phi = \frac{1}{\beta}\times -\frac{1}{\frac{1}{\beta}-t} e^{-y(1/\beta-t)}\bigg{|}^{\infty}_{0} =-\frac{1}{1-\beta t}\times\left[e^{-\infty}-e^0\right]=-\frac{1}{1-\beta t}\times\left(-1\right)=\color{red}\boxed{\frac{1}{1-\beta t}}$

Does this make sense?

--Chris

Thread: moment generating function of exponential dist.

LinkBack

Thread Tools

Display

moment generating function of exponential dist.

Similar Math Help Forum Discussions

Moment generating function from p.d.f

Moment Generating Function

moment generating function

moment generating function

Moment generating function

Search Tags