Expectation of Square of Discrete Random Variable
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Theorem
Let $X$ be a discrete random variable whose probability generating function is $\map {\Pi_X} s$.
Then the square of the expectation of $X$ is given by the expression:
- $\expect {X^2} = \map { {\Pi_X}''} 1 + \map { {\Pi_X}'} 1$
where $\map { {\Pi_X}''} 1$ and $\map { {\Pi_X}'} 1$ denote the second and first derivative respectively of the PGF $\map {\Pi_X} s$ evaluated at $1$.
Proof
From Derivatives of Probability Generating Function at One:
- $\map { {\Pi_X}''} 1 = \expect {X \paren {X - 1} }$
and from Expectation of Discrete Random Variable from PGF:
- $\map { {\Pi_X}'} 1 = \expect X$
So:
\(\ds \expect {X^2}\) | \(=\) | \(\ds \expect {X \paren {X - 1} + X}\) | Algebra: $X \paren {X - 1} + X = X^2 - X + X$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect {X \paren {X - 1} } + \expect X\) | Expectation is Linear | |||||||||||
\(\ds \) | \(=\) | \(\ds \map { {\Pi_X}''} 1 + \map { {\Pi_X}'} 1\) | from above |
$\blacksquare$
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.3$: Moments: $(19)$