Exponential Probability Distribution Calculator

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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.
exponential probability distribution
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



More References and links

  1. Statistics Calculators, Solvers and Graphers
  2. Normal Probability Calculator
  3. Normal Distribution Problems with Solutions
  4. Elementary Statistics and Probability Tutorials and Problems

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