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?

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