Contents
Does moment generating function always exist?
The moment-generating function of a real-valued distribution does not always exist, unlike the characteristic function. There are relations between the behavior of the moment-generating function of a distribution and properties of the distribution, such as the existence of moments.
Why do we need 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!
Are there moment generating functions for random variables?
There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions.
Why does the moment generating function not converge?
It may be that the moment generating function does not exist, because some of the moments may be infinite (or may not have a definite value, due to integrability issues). Also, even if the moments are all finite and have definite values, the generating function may not converge for any value of t other than 0.
Which is the expectation of a moment generating function?
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:
Why is the moment generating function called statlect?
The moment generating function takes its name by the fact that it can be used to derive the moments of , as stated in the following proposition. Proposition If a random variable possesses a mgf , then the -th moment of , denoted by , exists and is finite for any .