[Graduate Studies - Actuarial Math] - Distribution/Density functions

[All_That_Chaz]

I haven't the foggiest idea how to go about solving these problems. Any assistance would be awesome.


[note: the "x" in Mx(t) is a subscript]
1) Let Mx(t) is the moment generating function of a random variable X. Let R(t) = ln(Mx(t)). Then prove that
i) R'(0) = µ
ii) R''(0) = σ^2
Find R(t) if X is a standard normal variate and show the above results.


2) Let f(x) = 2g(x)G(λx). -∞<X<∞. Where g(x) and G(x) are the density and distribution function of the standard normal distribution respectively.
Is f(x) a density function? If yes, find the mean of the distribution.

[MrRubix]

Too lazy to actually show the steps but you can derive the answers to part 1 by looking up the definitions of moment generating functions -- http://mathworld.wolfram.com/Moment-...gFunction.html in particular shows how easily Part 1 can be handled, here. I've personally seldom used moment-generating functions so I unfortunately can't help with much more past that (although if you eyeball it you can see how the terms would drop out and net you the mean when you differentiate the natural log function, which is the derivative of x over x, so from there it's pretty straightforward. The variance (sigma-squared) portion is also defined here and can further be shown with the E(X^2)-E(X)^2 definition).

Part 2 I can tell you just from looking at it that it's a skew-uniform distribution

[Patashu]

oh mrrubix
solving dat grad statistics

[All_That_Chaz]

Well, I didn't really expect anybody to help beyond him. Although aperson surprised me in prochat.

I discussed the first one before class with other students and eventually figured it out. My prof told me to refer to a normal distribution for the second one and he gave everyone another week to do it after giving a couple hints involving double integrals and polar coordinates. He said he just gave us that problem to **** with us. I am not amused.