Moment generating function

• How to Find the Expected Value of an Exponential Function
  Find E[e^X] by integrating against X's distribution. For beta and gamma random variables, standard formulas give the result and its conditions.
• Why Can’t I Compute the Expected Value from a Uniform MGF?
  For X uniform on [a,b], the MGF has an apparent 0/0 at t=0; extend it by continuity, then differentiate to get E[X]=(a+b)/2.
• What Is the MGF of the Negative Binomial Distribution?
  For the failures-before-rth-success convention, the negative binomial MGF is [p/(1−(1−p)e^t)]^r; its domain and other conventions are explained.
• What Is the MGF of a Uniform Distribution?
  For a continuous uniform random variable on [a,b], the moment-generating function is (e^{bt}-e^{at})/[t(b-a)], with value 1 at t=0.