I'm trying to work out a practice problem but it seems like the way I'm doing it is so off track.
The goal is to find u'2 or the second moment generating function evaluated at t=0 for the random variable x with probability density
x/2 for 0<x<2
0 elsewhere
I started with 1/2*int(x*e^(tx),x, 0, 2)
(integrate x from 0 to 2) using the integration by parts I get this long integration of
.5[1/t*e^(tx)-1/t[1/te^(tx)]
Is the way I'm doing it even right up until this part? If it isn't how should I be doing it, and if it is, do I just take the derivative twice here with respect to t?
More generally, I think my question and confusion lies with this: If the probability density is given to be continuous like the one above but only from say 0 to 2 or 1 to infinity. Am I still integrating e^(tx)f(x) from 0 to infinity when I try to find the mgf? or am I only integrating from the defined 0 to 2 or 1 to infinity?
thanks a lot to anyone who can make this subject just a little simpler for me!
yes, thank you very much! This does wonders to solve a lot of the confusion I was having about the proper way to evaluate mgfs if the function's not defined as x>0. Now that I think about it, it makes so much sense, sense a lot of the problems were defined as x>0 so I've been integrating from 0 to infinity, it should make sense that it just carries through when its something like x>1
Edit: Follow up question should the X always disappear after finding the mgf? There's no way that you can get an mgf where the X still exists right? Otherwise you wouldn't be able to evaluate the derviatives of the mgfs at t?
Follow up question should the X always disappear after finding the mgf? There's no way that you can get an mgf where the X still exists right? Otherwise you wouldn't be able to evaluate the derviatives of the mgfs at t?