Probability Density Function of Exponential Distribution

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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.




Sources