for any K, and so the mgf is infinite for all t>0.

What Cannot be a moment generating function?

This seemingly weird function is actually quite useful in computing moments of random variables. where M′X(t) M X ′ ( t ) is the first derivative of the MGF of X with respect to t . Therefore, any function g(t) cannot be an MGF unless g(0)=1 g ( 0 ) = 1 .

Can moment generating function be zero?

Important properties

Moment generating functions are positive and log-convex, with M(0) = 1. may not exist. The log-normal distribution is an example of when this occurs.