Expectation of Exponential Distribution
Theorem
Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$
Then the expectation of $X$ is given by:
- $\expect X = \beta$
Proof 1
The expectation of a continuous random variable $X$ with sample space $\Omega_X$ is given by:
- $\ds \expect X := \int_{x \mathop \in \Omega_X} x \map {f_X} x \rd x$
where $f_X$ is the probability density function of $X$.
For the exponential distribution:
- $\Omega_X = \hointr 0 \infty$
From Probability Density Function of Exponential Distribution:
- $\ds \map {f_X} x = \frac 1 \beta \map \exp {- \frac x \beta}$
So:
- $\ds \expect X = \int_0^\infty x \frac 1 \beta \map \exp {- \frac x \beta} \rd x$
where $\exp$ is the exponential function.
Substituting $u = \dfrac x \beta$, we have:
- $\ds \expect X = \beta \int_0^\infty u \map \exp {-u} \rd u$
The integral evaluates to:
- $\ds \expect X = \bigintlimits {-\beta \paren {u + 1} \map \exp {-u} } 0 \infty$
So:
- $\ds \expect X = \beta - \beta \lim_{u \mathop \to \infty} \frac {u + 1} {\exp u}$
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By Limit at Infinity of Polynomial over Complex Exponential, it follows that this limit is zero, so that:
- $\expect X = \beta$
$\blacksquare$
Proof 2
From Moment Generating Function of Exponential Distribution, the moment generating function $M_X$ of $X$, is given by:
- $\map {M_X} t = \dfrac 1 {1 - \beta t}$
By Moment in terms of Moment Generating Function:
- $\expect X = \map {M_X'} 0$
We have:
\(\ds \map {M_X'} t\) | \(=\) | \(\ds \map {\frac \d {\d t} } {\frac 1 {1 - \beta t} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\beta} {-1} \frac 1 {\paren {1 - \beta t}^2}\) | Chain Rule for Derivatives, Derivative of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \beta {\paren {1 - \beta t}^2}\) |
Setting $t = 0$ gives:
\(\ds \expect X\) | \(=\) | \(\ds \frac \beta {\paren {1 - 0 \beta}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \beta\) |
$\blacksquare$
Also see
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): exponential distribution
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): exponential distribution
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $13$: Probability distributions
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions