PGF of Sum of Random Number of Independent Discrete Random Variables
Jump to navigation
Jump to search
Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let:
- $N, X_1, X_2, \ldots$
be independent discrete random variables such that the $X$'s have the same probability distribution.
Let:
Let:
- $Z = X_1 + X_2 + \ldots + X_N$
Then:
- $\map {\Pi_Z} s = \map {\Pi_N} {\map {\Pi_X} s}$
Proof
\(\ds \map {\Pi_Z} s\) | \(=\) | \(\ds \expect {s^{X_1 + X_2 + \cdots + X_N} }\) | Definition of Probability Generating Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop \ge 0} \expect {s^{X_1 + X_2 + \cdots + X_N} \mid N = n} \map \Pr {N = n}\) | Total Expectation Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop \ge 0} \expect {s^{X_1 + X_2 + \cdots + X_n} } \map \Pr {N = n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop \ge 0} \map {\Pi_X} s^n \map \Pr {N = n}\) | PGF of Sum of Independent Discrete Random Variables | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\Pi_N} {\map {\Pi_X} s}\) | Definition of $\map {\Pi_N} s$ |
$\blacksquare$
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.4$: Sums of independent random variables: Theorem $4 \text{D}$