Probability theory and stochastic processes Proposition

Generating function of a sum of independent random variables

Builds on Probability-generating function Expectation of a product of independent random variables Independence of random variables

Statement

Let $X_1, \dots, X_n$ be independent random variables with values in $\{0, 1, 2, \dots\}$ , and let $S = X_1 + \dots + X_n$ . Then for each complex $z$ with $|z| \le 1$ , the probability-generating functions satisfy $G_S(z) = G_{X_1}(z) \, G_{X_2}(z) \cdots G_{X_n}(z).$

Remarks

Proof: $z^S = z^{X_1} \cdots z^{X_n}$ . The factors are independent, so the product rule for expectations applies. Example: a sum of $n$ independent Bernoulli( $p$ ) variables has the generating function $(1 - p + pz)^n$ . This is the generating function of $\operatorname{Bin}(n, p)$ . For a random number $N$ of i.i.d. terms $X_i$ with $N$ independent of the $X_i$ , the sum $S_N = X_1 + \dots + X_N$ has $G_{S_N}(z) = G_N(G_{X_1}(z))$ .

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Generating function of a sum of independent random variables statement

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Sum of independent Poisson random variables

Tag	`0REMT01N36`
URI	`m2://0REMT01N36`
Aliases	`pgf-of-sum-of-independent-random-variables`
Kind	statement, proposition
Writing	W0 stub
MSC	60E05
Status	This copy of the site has no build data. Run `m2 status 0REMT01N36`.
Uses	Probability-generating function, Expectation of a product of independent random variables, Independence of random variables

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Sum of independent Poisson random variables