Question

Question: Find the moment generating function of having the density function f(x) = 26 27 x>0. Also find the variance of x.​

Answer 1

Answer:

How can i find the mean and variance by first getting the moment generating function and using it to find them.

The key here is to have a bit of an understanding of the MGF.

Suppose X is a random variable with MGF MX(t). By definition,

MX(t):=E[etX].

But, let's think about this for a moment. In terms of power series,

etX=∑n=0∞(tX)nn!=1+tX+12t2X2+16t3X3⋯

Computing the expectation (and sweeping a couple of important details under the rug) gives

MX(t)=E[etX]=1+E[X]t+12E[X2]t2+16E[X3]t3+⋯