Suppose two random variables and have a discrete joint distribution , for some real constant C. Suppose we want to find the moment-generating function for this distribution when and .
By definition, the MGF of this distribution is , where the function represents the expected value. Since its a discrete distribution, we write as:
… by the definition of the binomial coefficient.
Multiply through by and rearrange terms to get:
The inner summation reduces nicely by the binomial formula and we get:
By the Taylor series expansion for the exponential function, we can rewrite this as:
Isn’t multiplying by one such an awesome little trick?