Expectation of Geometric Distribution/Formulation 2
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Theorem
Let $X$ be a discrete random variable with the geometric distribution with parameter $p$ for some $0 < p < 1$.
- $\map X \Omega = \set {0, 1, 2, \ldots} = \N$
- $\map \Pr {X = k} = p \paren {1 - p}^k$
Then the expectation of $X$ is given by:
- $\map E X = \dfrac {1-p} p$
Proof 1
From the definition of expectation:
- $\ds \expect X = \sum_{x \mathop \in \Omega_X} x \map \Pr {X = x}$
By definition of geometric distribution:
- $\ds \expect X = \sum_{k \mathop \in \Omega_X} k p \paren {1 - p}^k$
Let $q = 1 - p$:
\(\ds \expect X\) | \(=\) | \(\ds p \sum_{k \mathop \ge 0} k q^k\) | as $\Omega_X = \N$ | |||||||||||
\(\ds \) | \(=\) | \(\ds p \sum_{k \mathop \ge 1} k q^k\) | as the $k = 0$ term is zero | |||||||||||
\(\ds \) | \(=\) | \(\ds pq \sum_{k \mathop \ge 1} k q^{k - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p q \frac 1 {\paren {1 - q}^2}\) | Derivative of Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {p q} {p^2}\) | as $q = 1 - p$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {1 - p} p\) |
$\blacksquare$
Proof 2
By Moment Generating Function of Geometric Distribution, the moment generating function of $X$ is given by:
- $\map {M_X} t = \dfrac p {1 - \paren {1 - p} e^t}$
for $t < -\map \ln {1 - p}$, and is undefined otherwise.
From Moment in terms of Moment Generating Function:
- $\expect X = \map { {M_X}'} 0$
From Moment Generating Function of Geometric Distribution: First Moment:
- $\map { {M_X}'} t = \dfrac {p \paren {1 - p} e^t } {\paren {1 - \paren {1 - p} e^t}^2 }$
Hence setting $t = 0$:
\(\ds \map { {M_X}'} 0\) | \(=\) | \(\ds \dfrac {p \paren {1 - p} } {\paren {1 - \paren {1 - p} }^2 }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {p \paren {1 - p} } {p^2 }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1 - p} p\) |
$\blacksquare$
Also presented as
The Expectation of Geometric Distribution is also presented in the form:
- $\var X = \dfrac q p$
where $q$ has been defined conventionally as $q = 1 - p$.
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): geometric distribution