Probability Density Function of Exponential Distribution

Theorem

Let $X$ be a continuous random variable with the exponential distribution with parameter $\beta$.

Then the probability density function of $X$ is given by:

$\map {f_X} x = \begin{cases} \dfrac 1 \beta e^{-\frac x \beta} & : x \ge 0 \\ 0 & : \text{otherwise} \end{cases}$

By definition of exponential distribution:

$\map {F_X} \Omega = \R_{\ge 0}$

$\map \Pr {X < x} = 1 - e^{-\frac x \beta}$

where $0 < \beta$.

By definition of probability density function:

$\forall x \in \R: \map {f_X} x = \begin {cases} \map {F_X'} x & : x \in \Sigma \\ 0 & : x \notin \Sigma \end {cases}$

where $\map {F_X'} x$ denotes the derivative of $F_X$ at $x$.

Then:

$\ds f_X$

$=$

$\ds \map {\dfrac \d {\d x} } {1 - e^{-\frac x \beta} }$

$\ds $

$=$

$\ds \dfrac 1 \beta e^{-\frac x \beta}$

Hence the result.

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): exponential distribution
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): frequency function
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): exponential distribution
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): frequency function
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