Why are moment generating functions useful in determining the distribution of a random variable?

Why are moment generating functions useful in determining the distribution of a random variable?

Moment generating functions are useful for several reasons, one of which is their application to analysis of sums of random variables. The nth central moment of X is defined to be E[(X−EX)n]. For example, the first moment is the expected value E[X]. The second central moment is the variance of X.

What is moment generating function of chi-square distribution?

I’m tasked with deriving the MGF of a χ2 random variable. I think the way to do is is by using the fact that Σmj=1Z2j is a χ2 R.V. and that MGF of a sum is the product of the MGFs of the individual terms.

Why do we find moment generating function?

MGF encodes all the moments of a random variable into a single function from which they can be extracted again later. A probability distribution is uniquely determined by its MGF. If two random variables have the same MGF, then they must have the same distribution. And we can detect those using MGF!

How to find the moment generating function of chi-squared distribution-cross?

I think the way to do is is by using the fact that Σmj = 1Z2j is a χ2 R.V. and that MGF of a sum is the product of the MGFs of the individual terms. Although that may not be right and it may be E(etX) way. I don’t need it solved really just need to get down the track a little further.

Why is the moment generating function of X important?

The moment generating function of X is Besides helping to find moments, the moment generating function has an important property often called the uniqueness property. The uniqueness property means that, if the mgf exists for a random variable, then there one and only one distribution associated with that mgf.

How to find the variance of a moment generating function?

We can solve these in a couple of ways. We can use the knowledge that M ′ ( 0) = E ( Y) and M ′ ′ ( 0) = E ( Y 2). Then we can find variance by using V a r ( Y) = E ( Y 2) − E ( Y) 2. This is left as an exercise below. We can recognize that this is a moment generating function for a Geometric random variable with p = 1 4.

How to find the distribution of a random variable?

Recall that the moment generating function: uniquely defines the distribution of a random variable. That is, if you can show that the moment generating function of X ¯ is the same as some known moment-generating function, then X ¯ follows the same distribution. So, one strategy to finding the distribution of a function of random variables is: