1. ## MGF Problem

Hi

Can someone help with the pointing me in right direction with the following

"Use the MGF to compute the mean and variance of the distribution of x which has the following probability distribution

f(x) = x+1/14 if x=1,2,3,4

Many thanks

2. Originally Posted by statsnewbie
Hi

Can someone help with the pointing me in right direction with the following

"Use the MGF to compute the mean and variance of the distribution of x which has the following probability distribution

f(x) = x+1/14 if x=1,2,3,4

Many thanks
The mass density function is $\displaystyle f(x) = \frac{x + 1}{14}$ where x = 1, 2, 3, 4.

By definition, the moment generating function m(t) for a random variable X is

$\displaystyle E\left( e^{tX}\right) = \Pr(X = x_1) \, e^{tx_1} + \Pr(X = x_2) \, e^{tx_2} + \Pr(X = x_3) \, e^{tx_3} + ......$.

$\displaystyle m(t) = \Pr(X = 1) \, e^{t} + \Pr(X = 2) \, e^{2 t} + \Pr(X = 3) \, e^{3 t} + \Pr(X = 4) \, e^{4 t}$

$\displaystyle = \frac{1}{14} \left( 2 \, e^{t} + 3 \, e^{2 t} + 4 \, e^{3 t} + 5 \, e^{4 t}\right)$.

By definition: $\displaystyle \mu = E(X) = \frac{dm}{dt}$ evaluated at t = 0:

............

By definition: $\displaystyle E(X^2) = \frac{d^2 m}{d t^2}$ evaluated at t = 0:

.............

Formula: $\displaystyle Var(X) = E(X^2) - \mu^2$:

.............

I've left the details for you to fill in.

3. ## MGF Problem

Thanks

One further question.

If t=0 does that mean that each element of (e) evaluates to zero or have I missed something.

4. Originally Posted by statsnewbie
Thanks

One further question.

If t=0 does that mean that each element of (e) evaluates to zero or have I missed something.
?
You find the appropriate derivatives and then substitute t = 0. You do know that e^0 = 1 I hope .....