depending on the moment generating function of the random variable X : M(t)=e^(t^2 ) :find

1. p(x=2)

2.find the mean of X

3. FIND THE VARIANCE of -2x

4.find the the second moment about the origin

i don't know how i starting ??

Results 1 to 10 of 10

- Aug 6th 2008, 10:36 AM #1

- Joined
- Aug 2008
- Posts
- 172

- Aug 6th 2008, 07:15 PM #2
You're expected to know the following formula: $\displaystyle E\left( X^n \right) = \left. \frac{d^n M}{d t^n}\right| _{t=0}$

2. $\displaystyle \mu = E(X) = \left. \frac{d M}{d t}\right| _{t=0} = \, ....$

4. $\displaystyle E \left( X^2\right) = \left. \frac{d^2 M}{d t^2}\right| _{t=0} = \, ....$

3. $\displaystyle Var(-2X) = (-2)^2 Var(X) = 4 Var(X)$ and $\displaystyle Var(X) = E \left( X^2\right) - \mu^2$.

Now substitute the results from 2. and 4.

1. If X is a continuous random variable then Pr(X = a) = 0.

I have more to say and when I have the time I'll say it.

- Aug 6th 2008, 07:51 PM #3
Note that the normal distribution with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$ has the moment generating function $\displaystyle e^{\mu t + \frac{t^2 \sigma^2}{2}}$.

Compare this with the given moment generating function and you see that X must follow a normal distribution with $\displaystyle \mu = 0$ and variance $\displaystyle \sigma^2 = 2$.

This gives you a way of checking the answers to 2. and 3.

It also lets you calculate Pr(a < X < b) were it to be asked.

- Aug 7th 2008, 03:51 AM #4

- Joined
- Aug 2008
- Posts
- 172

- Aug 7th 2008, 04:00 AM #5

- Joined
- Aug 2008
- Posts
- 172

- Aug 7th 2008, 04:09 AM #6
If you showed your working I could identify the mistake(s) you made in getting Var(X) = 0.

$\displaystyle \frac{d^2 M}{dt^2} = 2 e^{t^2} + 4t^2 e^{t^2}$. Substitute t = 0 and you get $\displaystyle E\left(X^2\right) = \left.\frac{d^2 M}{dt^2} \right|_{t=0} = 2$.

Therefore $\displaystyle \sigma^2 = Var(X) = E(X^2) - \mu^2 = 2 - 0 = 2$, as expected.

Therefore Var(-2X) = 4 Var(X) = 8.

Feel free to repeat post #4 now lol!

- Aug 7th 2008, 06:13 AM #7

- Joined
- Aug 2008
- Posts
- 172

i'm so sorry i solve it before i open my mail and see the solution >>>

i forgot that the derivative for 2te^(t^2 ) is equal to "derivative for two multiply function " so the answer is false >>>>>

but i solve it >>>>>then i verify from your saying

thank you>>>>you're a nice man

- Aug 7th 2008, 06:16 AM #8

- Joined
- Aug 2008
- Posts
- 172

- Aug 7th 2008, 03:28 PM #9

- Aug 8th 2008, 02:09 AM #10

- Joined
- Aug 2008
- Posts
- 172