moment generating function

**logitech** · May 3rd 2009, 01:47 PM

If the moment generating function of X is $M(t)=\frac{1}{1-3t}$ , $t<\frac{1}{3}$

A. Find E(X)
B. Find Var(X)
C. P(6.1 < x < 6.7)

Thank you in advance for your help!

**Moo** · May 3rd 2009, 02:10 PM

Hello,

Originally Posted by logitech

If the moment generating function of X is $M(t)=\frac{1}{1-3t}$ , $t<\frac{1}{3}$

A. Find E(X)
B. Find Var(X)
C. P(6.1 < x < 6.7)

Thank you in advance for your help!

By definition, $E(X^n)=M^{(n)}(0)$ , where $M^{(n)}$ denotes the n-th derivative of the mgf.

Specifically, $E(X)=M'(0)$ and $E(X^2)=M''(0)$

For Var(X), recall the formula I put before : E(X²)-[E(X)]²

For question C., I don't know... Your mgf is the mgf of an exponential distribution with parameter 1/3, which would finish the question, by using the cumulative distribution function.

I'm looking for a way without "recognizing" the mgf, but it looks like there isn't any.

**logitech** · May 3rd 2009, 02:28 PM

This one is still difficult for me to understand. Thank you for trying to help me understand this one. I will keep working on it. I wish you could be my tutor! Taking an online class basically means there is no teacher, just a textbook and a list of work to do.

**matheagle** · May 3rd 2009, 02:46 PM

MGF's are unique
This is a gamma with $\alpha=1$ and $\beta=3$

Hence the mean is $\alpha\beta=3$
and the variance is $\alpha\beta^2=9$
and yes, ms mathbeagle, now you can integrate the density to obtain the probabilities.

**mr fantastic** · May 16th 2009, 03:02 AM

Originally Posted by logitech

If the moment generating function of X is $M(t)=\frac{1}{1-3t}$ , $t<\frac{1}{3}$

A. Find E(X)
B. Find Var(X)
C. P(6.1 < x < 6.7)

Thank you in advance for your help!

See this thread also: http://www.mathhelpforum.com/math-he...enerating.html

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