1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Mgf of a random variable with added constant

  1. Nov 3, 2011 #1
    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:

    [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

    User Avatar
    Science Advisor
    Homework Helper

    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
     
  7. Nov 3, 2011 #6
    Checked that and it was another mistake I was making. Thank you for pointing this out!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Mgf of a random variable with added constant
  1. Random variable (Replies: 1)

  2. Variable constant? (Replies: 3)

  3. Random variable (Replies: 2)

Loading...