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Mgf of a random variable with added constant 
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#1
Nov311, 12:40 PM

P: 10

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
Nov311, 02:24 PM

P: 828

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? 


#3
Nov311, 03:02 PM

P: 10




#4
Nov311, 03:10 PM

P: 828

Mgf of a random variable with added constant
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
Nov311, 04:53 PM

Sci Advisor
HW Helper
Thanks
P: 4,958

RGV 


#6
Nov311, 05:09 PM

P: 10




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