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

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:

Hence the result.