Is probability generating function the same as Moment generating function?

The probability generating function is usually used for (nonnegative) integer valued random variables, but is really only a repackaging of the moment generating function. So the two contains the same information.

What is the difference between MGF and PGF?

Specifically, I understand that a MGF is used to calculate the moments of either a discrete or continuous distribution and then build that distribution by summing these moments (similar to how a Taylor Series works). A PGF is a more general version of a MGF but can only be applied to discrete distributions.

What is moment generating function in probability?

The moment-generating function is the expectation of a function of the random variable, it can be written as: For a discrete probability mass function, For a continuous probability density function, In the general case: , using the Riemann–Stieltjes integral, and where is the cumulative distribution function.

Why do we need moment generating function?

Moments provide a way to specify a distribution. 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.

Why is it beneficial to use moment generating functions?

Moment generating functions are useful for several reasons, one of which is their application to analysis of sums of random variables. The nth moment of a random variable X is defined to be E[Xn]. 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].

Why are moment generating functions important?

The moment generating function has great practical relevance because: it can be used to easily derive moments; its derivatives at zero are equal to the moments of the random variable; a probability distribution is uniquely determined by its mgf.