# 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