Probability Density Function of Exponential Distribution
Jump to navigation
Jump to search
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:
- $\map {f_X} x = \begin{cases} \dfrac 1 \beta e^{-\frac x \beta} & : x \ge 0 \\ 0 & : \text{otherwise} \end{cases}$
Proof
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.
This needs considerable tedious hard slog to complete it. In particular: Don't have the patience to resolve all the fiddle-faddle concerning the various notations, which are all over the place and far from being consistent To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 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): 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