# Mgf of a random variable with added constant

1. Nov 3, 2011

### WantToBeSmart

Hey,

I have a pdf of a random variable Z given. I am being asked to calculate what the moment generating function of a r.v Y= Z + c will be where c is a constant in ℝ

I tried to calculate it in the following way:

$$\int^∞_0 e^{(z+c)t} f(z+c)dz$$ where $$f(z)$$ is an exponential pdf with parameter λ.

but it proved to be an unsuccessful method. Could anyone please show me the right direction? I know I could use Jacobian transformation but I'm sure there is an easier method.

2. Nov 3, 2011

### Robert1986

I wouldn't even mess around with the integral. Here is something I would try:

$Y = Z + c$ where $Z ~ exp(\lambda)$ and c is a constant. Then,

$E[e^{tY}] = E[e^{t(Z+c)}]$

Now do you see what you might be able to do?

3. Nov 3, 2011

### WantToBeSmart

I think it definitely solves this problem! Now I can proceed with the rest of the exercise. Thank you Robert!

4. Nov 3, 2011

### Robert1986

You're most certainly welcome.

As a side note, this sort of thing is a rather valuable technique in prob/stat. That is, if you want to know about a certain RV, or a certain expectation, lots of times it is best to work it into some form you already know.

5. Nov 3, 2011

### Ray Vickson

Of course, you would have gotten the same result had you used the correct f(z) dz in your integration, instead of your _incorrect_ f(z+c) dz.

RGV

6. Nov 3, 2011

### WantToBeSmart

Checked that and it was another mistake I was making. Thank you for pointing this out!