# 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 Distribution

A continuous exponential probability distribution 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