PGF of Sum of Independent Discrete Random Variables

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Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ and $Y$ be independent discrete random variables on $\struct {\Omega, \Sigma, \Pr}$.

Let $Z$ be a discrete random variable such that $Z = X + Y$.

Then:

$\map {\Pi_Z} s = \map {\Pi_X} s \, \map {\Pi_Y} s$

where $\map {\Pi_Z} s$ is the probability generating function of $Z$.


General Result

Let:

$Z = X_1 + X_2 + \cdots + X_n$

where each of $X_1, X_2, \ldots, X_n$ are independent discrete random variables with PGFs $\map {\Pi_{X_1} } s, \map {\Pi_{X_2} } s, \ldots, \map {\Pi_{X_n} } s$.


Then:

$\ds \map {\Pi_Z} s = \prod_{j \mathop = 1}^n \map {\Pi_{X_j} } s$


Proof

\(\ds \map {\Pi_Z} s\) \(=\) \(\ds \expect {s^Z}\) Definition of Probability Generating Function
\(\ds \) \(=\) \(\ds \expect {s^{X + Y} }\) Definition of $Z$ (see above)
\(\ds \) \(=\) \(\ds \expect {s^X s^Y}\) Exponential of Sum
\(\ds \) \(=\) \(\ds \expect {s^X} \expect {s^Y}\) Condition for Independence from Product of Expectations
\(\ds \) \(=\) \(\ds \map {\Pi_X} s \, \map {\Pi_Y} s\) Definition of Probability Generating Function

$\blacksquare$


Sources

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