1. ## moment generating function

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)

2. 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)

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.

3. 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.

4. 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.

5. 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)