# Moment Generating Function problem

Glass
I'm given the probability density function:
f(x) = 3x^2 for x in [0, 1]
f(x) = 0 elsewhere
I want to find E[X^2] which is easy if I use the integral definition (I got 3/5). Yet, when I try and do this using Moment Generating Function (mgf) I cannot seem to get the same answer (in fact I get an unbounded answer). I'm wondering why this is the case. Any help?

Homework Helper
Gold Member
Well as you guessed, that's not normal. So there has to be something quite not right with one of your calculations. But nobody can help you find what is the problem if we can't see your calculations.

Glass
From the definition of mgf I have:
$$M(t) = \int_{0}^{1}e^{xt}3x^2dx$$
Then after some integration by parts (or using a symbolic integrator) I get:
$$M(t) = e^t(\frac{3}{t} - \frac{6}{t^2} + \frac{6}{t^3}) - \frac{6}{t^3}$$
Then if we differentiate that twice (or even once or zero times) there are still 1/t (or some power of 1/t) terms in there.

$$M_X(t)=\int_{0}^{1}e^{tx}dx=\frac{1}{t}(e^t-1)$$