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Mgf of a random variable with added constant

  1. Nov 3, 2011 #1

    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:

    [tex] \int^∞_0 e^{(z+c)t} f(z+c)dz[/tex] where [tex] f(z) [/tex] 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.

    Thank you in advance!
  2. jcsd
  3. Nov 3, 2011 #2
    I wouldn't even mess around with the integral. Here is something I would try:

    [itex] Y = Z + c [/itex] where [itex] Z ~ exp(\lambda) [/itex] and c is a constant. Then,

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

    Now do you see what you might be able to do?
  4. Nov 3, 2011 #3
    I think it definitely solves this problem! Now I can proceed with the rest of the exercise. Thank you Robert!
  5. Nov 3, 2011 #4
    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.
  6. Nov 3, 2011 #5

    Ray Vickson

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    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.

  7. Nov 3, 2011 #6
    Checked that and it was another mistake I was making. Thank you for pointing this out!
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