Probability generating function

The probability generating function is a generating function for the probability mass function of a $ℕ 0$ -valued discrete random variable $𝑋 : 𝜉 \to ℕ 0$ defined by #m/def/prob

𝑔 𝑋 (𝑡) : = \infty \sum 𝑥 = 0 𝑝 𝑋 (𝑥) 𝑡 𝑥 = 𝔼 [𝑡 𝑋]

by the Law of the unconscious statistician. This is well-defined as a convergent function $𝑔 𝑋 : [- 1, 1] \to [- 1, 1]$ .

Properties

If the Moment-generating function exists, for $𝑡 < 0$ $𝑔 𝑋 (𝑡) = 𝔼 [𝑡 𝑋] = 𝔼 [e 𝑋 l n 𝑡] = 𝑀 𝑋 (l n 𝑡)$
$ℙ [𝑋 = 𝑥] = 𝑔 (𝑥) 𝑋 (0) 𝑥!$
Let $𝑋, 𝑌 : 𝜉 \to ℕ 0$ be independent random variables. Then $𝑔 𝜆 𝑋 + 𝜇 𝑌 (𝑡) = 𝑔 𝑋 (𝑡 𝜆) + 𝑔 𝑌 (𝑡 𝜇)$

#state/tidy | #lang/en | #SemBr