Properties of Probability Generating Function

Theorem
Let $$X$$ be a discrete random variable whose probability generating function is $$\Pi \left({s}\right)$$.

Then $$\Pi \left({s}\right)$$ has the following properties:

PGF defines PMF

 * The coefficient of $$s^x$$ in $$\Pi \left({s}\right)$$ is $$p_X \left({x}\right)$$, where $$p_X$$ is the probability mass function of $$X$$.

PGF determines Distribution
The probability generating function uniquely determines a discrete probability distribution, and vice versa.

PGF of 0

 * $$\Pi \left({0}\right) = p_X \left({0}\right)$$

PGF of 1

 * $$\Pi \left({1}\right) = 1$$

Proof that PGF defines PMF
By definition of probability generating function:
 * $$\Pi \left({s}\right) = p_X \left({0}\right) + p_X \left({1}\right) s + p_X \left({2}\right)s^2 + \cdots$$

Hence the truth of the assertion is clear.

Proof that PGF determines Distribution
We note that the coefficient of $$s^x$$ in the PGF is $$p_X \left({x}\right)$$ for each $$x$$.

So the probability mass function determines the PGF uniquely, and vice versa.

Proof of PGF of 0
$$ $$ $$

Proof of PGF of 1
$$ $$ $$ $$