Properties of Probability Generating Function

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

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

PGF defines PMF

 * The coefficient of $$s^x$$ in $$\Pi_X \left({s}\right)$$ 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.

That is:
 * $$\Pi_X \left({s}\right) = \Pi_Y \left({s}\right) \iff \forall k \in \N: p_X \left({k}\right) = p_Y \left({k}\right)$$

PGF of 0

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

PGF of 1

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

Proof that PGF defines PMF
It is clear that:
 * $$\forall k \in \N: p_X \left({k}\right) = p_Y \left({k}\right) \implies \Pi_X \left({s}\right) = \Pi_Y \left({s}\right)$$

from the method of construction of the probability generating function.

So, suppose that $$\Pi_X \left({s}\right) = \Pi_Y \left({s}\right)$$.

By definition of probability generating function:

$$ $$

If $$\Pi_X \left({s}\right) = \Pi_Y \left({s}\right)$$ then it is clear that these two power series have identical coefficients.

Hence the result.

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
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Proof of PGF of 1
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