I'm trying to make sense of a problem we saw in my probability class. mr. fantastic helped me dust off my calculus skills for the integration part but now I'm having trouble understanding the very last part. Here's the problem and its solution:
The probability density of a continuous random variable Y is given by
Prove that the moment generating function, m(t), of Y is given by
Can someone please explain
to me? I'm confused by the very end. If k is the one being taken to then why should the outcome depend on t? Please explain how we get and .
When you say "which really doesn't make sense", what are you referring to? Are you suggesting that there ought to be a third case for ?If , then and the limit would diverge for , making and if . [which really doesn't make sense].
BTW, I should have noticed the correction you made myself... The question actually states "for t < "