Variance of Shifted Geometric Distribution

Theorem

Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$.

Then the variance of $X$ is given by:

$\displaystyle \operatorname{var} \left({X}\right) = \frac {1-p} {p^2}$

Proof

From the definition of Variance as Expectation of Square minus Square of Expectation:

$\operatorname{var} \left({X}\right) = E \left({X^2}\right) - \left({E \left({X}\right)}\right)^2$

From Expectation of Function of Discrete Random Variable:

$\displaystyle E \left({X^2}\right) = \sum_{x \in \operatorname{Im} \left({X}\right)} x^2 \Pr \left({X = x}\right)$

To simplify the algebra a bit, let $q = 1 - p$, so $p+q = 1$.

Thus:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle E \left({X^2}\right)\)	\(=\)	\(\displaystyle \sum_{k \ge 0} k^2 p q^{k - 1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of shifted geometric distribution, with $p + q = 1$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{k \ge 1} k^2 p q^{k - 1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	The term in $k=0$ is zero, so we change the limits
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{k \ge 1} k \left({k + 1}\right) p q^{k - 1} - \sum_{k \ge 1} k p q^{k - 1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	splitting sum up into two
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{k \ge 1} k \left({k + 1}\right) p q^{k - 1} - \frac 1 p\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Second term is Expectation of Shifted Geometric Distribution
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle p \frac 2 {\left({1-q}\right)^3} - \frac 1 p\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from Derivative of Geometric Progression: Corollary
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 2 {p^2} - \frac 1 p\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	putting $p = 1-q$ back in and simplifying

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \operatorname{var} \left({X}\right)\)	\(=\)	\(\displaystyle E \left({X^2}\right) - \left({E \left({X}\right)}\right)^2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 2 {p^2} - \frac 1 p - \frac 1 {p^2}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Expectation of Shifted Geometric Distribution: $E \left({X}\right) = \dfrac 1 p$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 1 {p^2} - \frac 1 p\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {1 - p} {p^2}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Proof 2

From Variance of Discrete Random Variable from P.G.F., we have:

$\operatorname{var} \left({X}\right) = \Pi''_X \left({1}\right) + \mu - \mu^2$

where $\mu = E \left({x}\right)$ is the expectation of $X$.

From the Probability Generating Function of Shifted Geometric Distribution, we have:

$\Pi_X \left({s}\right) = \dfrac {ps} {1 - qs}$

where $q = 1 - p$.

From Expectation of Shifted Geometric Distribution, we have:

$\mu = \dfrac 1 p$

From Derivatives of PGF of Shifted Geometric Distribution, we have:

$\Pi''_X \left({s}\right) = \dfrac {p q} {\left({1 - qs}\right)^3}$

Putting $s = 1$ using the formula $\Pi''_X \left({1}\right) + \mu - \mu^2$:

$\displaystyle \operatorname{var} \left({X}\right) = \frac {p q} {\left({1 - q}\right)^3} + \frac 1 p - \left({\frac 1 p}\right)^2$

and hence the result, after some algebra.

$\blacksquare$

Sources

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 2.4$: Example $24$, $\S 4.3$: Example $21$