Find the MGF of X

Question

If $f(x) = (k + 1)x^2$ for $0 < x < 1$. Find the moment generating function of $X$.

Do i do the integral from o to 1 of the above function?

Answer 1

The moment generation function $G_X(t)$ is the expectation of $e^{Xt}$. This is $$\int_0^1 e^{xt}(k+1)(x^2)\,dx.$$ Note that you will first want to find $k$.

Answer 2

You first need to determine the $k$ by forcing your distribution to integrate to one over the given interval of $x$, i.e.$$ \int_0^1 \left( k +1 \right) x^2 dx = 1 $$ Solve for $k$ and then proceed by taking the expectation again over that same interval after having plugged in the value of $k$.

Note that all expectations are taken over the probability space of the density function and the MGF is a special case of an expectation.