# Exponential Probability Distribution Calculator

   

An online calculator that calculates the probabilities related to exponential probability distributions is presented. A second calculator solves the inverse problem: given $p$ such that $P(X \lt x_1) = p$, calculate $x_1$.

## Exponential Probability Ditribution

A continuous uniform probability ditribution has the probability density function of the form
$f(x,\lambda) = \begin{cases} \lambda e^{-\lambda x} \quad \text{for} \quad x \ge 0 \\ \\ 0 \quad \text{for} \quad x \lt 0 \\ \end{cases}$
where $\lambda \gt 0$ is called the rate of the distribution.
Graphs of exponential distributions, with different values of the rate $\lambda$ are shown below.

The probability that the random variable $X$ is less than $x_1$ is given by $\displaystyle P(X \lt x_1) = \int_{0}^{x_1} \lambda e^{-\lambda x} \; dx = 1 - e^{-\lambda x_1} \quad \text{for} \quad x_1 \ge 0$
The mean, variance and standard deviation of an exponential probability distribution, as defined above, are given by:
Mean = $\dfrac{1}{\lambda}$
Variance = $\dfrac{1}{\lambda^2}$
Standard Deviation = $\dfrac{1}{\lambda}$
We present two calculators.

## 1 - Find the probability $P(X \lt x_1)$ given $\lambda$ and $x_1$

$\lambda$ =     $x_1$ =

Decimal Places =

Output

## 2 - Inverse Problem: Find $x_1$ such that $P(X \lt x_1) = p$ given $\lambda$ and $p$:

$\lambda$ =     $p$ =

Output