Probability theory MOC

Probability-generating function

The probability generating function is a generating function for the probability mass function of a ℕ0-valued discrete random variable 𝑋 :𝜉 →ℕ0 defined by prob

𝑔𝑋(𝑡):=∞∑𝑥=0𝑝𝑋(𝑥)𝑡𝑥=𝔼⁡[𝑡𝑋]

by the Law of the unconscious statistician. This is well-defined as a convergent function 𝑔𝑋 :[ −1,1] →[ −1,1].

Properties

1. If the Moment-generating function exists, for 𝑡 <0

𝑔𝑋(𝑡)=𝔼⁡[𝑡𝑋]=𝔼⁡[e𝑋ln⁡𝑡]=𝑀𝑋(ln⁡𝑡)

ℙ[𝑋=𝑥]=𝑔(𝑥)𝑋(0)𝑥!

3. Let 𝑋,𝑌 :𝜉 →ℕ0 be independent random variables. Then

𝑔𝜆𝑋+𝜇𝑌(𝑡)=𝑔𝑋(𝑡𝜆)+𝑔𝑌(𝑡𝜇)

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