Moment-generating function

The moment-generating function $𝑀 𝑋 (⃗ 𝐭)$ of a real random variable or random vector $⃗ 𝐗 : 𝜉 \to ℝ 𝑘$ is defined as #m/def/prob

𝑀 𝑋 : (- 𝑎, 𝑎) 𝑘 \to ℝ ⃗ 𝐭 \mapsto 𝔼 [e ⃗ 𝐭 \cdot ⃗ 𝐗]

provided it is finite on some interval $(- 𝑎, 𝑎)$ around 0, otherwise the moment-generating function does not exist.

tip

To make sure a moment-generating function is valid, check that $𝑀 𝑋 (0) = 1$ .

If the moment-generating functions of two real random variables match in an arbitrarily small neighbourhood of 0, they must have the same distribution. #m/thm/prob Thus moment-generating function carries all relevant information about the distribution of $𝑋$ , and therefore provides an alternative to working with the probability density function and cumulative distribution function directly.

Relation to moments

Taking the Taylor expansion of $𝑀 𝑋$ it follows

𝑀 (𝑛) (0) = 𝔼 [𝑋 𝑛] = 𝑀 𝑛

the $𝑛$ th moment of $𝑋$ . #m/thm/prob Hence $𝑀 𝑋$ is a generating function for the moments of $𝑋$ .

Properties

$𝑀 𝑋 + 𝑌 (𝑡) = 𝑀 𝑋 𝑀 𝑌 (𝑡)$ for independently distributed $𝑋, 𝑌$
$𝑀 𝑎 + 𝑏 𝑋 (𝑡) = 𝑒 𝑎 𝑡 𝑀 𝑋 (𝑏 𝑡)$

Proof of 1–2

By definition, if it exists

𝑀 𝑋 + 𝑌 (𝑡) = 𝔼 [e 𝑡 𝑋 + 𝑡 𝑌] = 𝔼 [e 𝑡 𝑋 e 𝑡 𝑌] = 𝔼 [e 𝑡 𝑋] 𝔼 [e 𝑡 𝑌]

by ^P3, proving ^P1. Similarly

𝑀 𝑎 + 𝑏 𝑋 = 𝔼 [e 𝑡 (𝑎 + 𝑏 𝑋)] = e 𝑎 𝑡 𝔼 [e 𝑡 𝑏 𝑋] = e 𝑎 𝑡 𝑀 𝑋 (𝑏 𝑡)

proving ^P2.

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