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