Laplace distribution examples
Laplace Distribution Calculator
Use this calculator to find the probability density and cumulative probabilities for Laplace distribution with parameter $\mu$ and $\lambda$.
Laplace Distribution Calculator | |
---|---|
Location parameter $\mu$: | |
Scale parameter $\lambda$ | |
Value of x | |
Results | |
Probability density : f(x) | |
Probability X less than x: P(X < x) | |
Probability X greater than x: P(X > x) | |
How to calculate probabilities of Laplace Distribution?
Step 1 – Enter the location parameter $\mu$
Step 2 – Enter the scale parameter $\lambda$
Step 3 – Enter the value of $x$
Step 4 – Click on "Calculate" button to get Laplace distribution probabilities
Step 5 – Gives the output probability at $x$ for Laplace distribution
Step 6 – Gives the output cumulative probabilities for Laplace distribution
Definition of Laplace distribution
A continuous random variable $X$ is said to have a Laplace distribution, if its p.d.f. is given by
$$ \begin{align*} f(x;\mu, \lambda)&= \begin{cases} \frac{1}{2\lambda}e^{-\frac{|x-\mu|}{\lambda}}, & -\infty < x < \infty; \\ & -\infty < \mu < \infty, \lambda >0; \\ 0, & Otherwise. \end{cases} \end{align*} $$
Distribution Function of Laplace Distribution
Distribution function of $L(\mu,\lambda)$ distribution is
$$ \begin{align*} F(x) &= \begin{cases} \frac{1}{2}e^{\frac{(x-\mu)}{\lambda}}, & x < \mu; \\ &\\ 1-\frac{1}{2}e^{-\frac{(x-\mu)}{\lambda} }, & x\geq \mu; \end{cases} \end{align*} $$
Mean of Laplace Distribution
The mean of Laplace distribution is $E(X) = \mu$.
Variance of Laplace Distribution
The variance of Laplace distribution is $V(X) = 2\lambda^2$.
Laplace Distribution Example 1
A random variable $X$ follows a Laplace distribution with parameter $\mu =5$ and $\lambda=2$.
Find the probability that
a. $X$ is less than 1,
b. $X$ is less than 6,
c. $X$ is between 6 and 10,
d. $X$ is greater than 3.5.
Solution
a. The probability that $X$ is less than $1$ is
$$ \begin{aligned} P(X \leq 1) &=F(1)\\ &=\frac{1}{2}e^{\dfrac{(1-5)}{2}}\\ &\qquad (\because 1< mu)\\ &=\frac{1}{2}e^{\dfrac{(1-5)}{2}}\\ &= 0.0677 \end{aligned} $$
b. The probability that $X$ is less than $6$ is
$$ \begin{aligned} P(X \leq 6) &=F(6)\\ &=1-\frac{1}{2}e^{\dfrac{-(6-5)}{2}}\\ &\qquad (\because 6> mu)\\ &=1-\frac{1}{2}e^{\dfrac{-(6-5)}{2}}\\ &= 1-0.3033\\ &= 0.6967 \end{aligned} $$
c. The probability that $X$ is between $6$ and $10$ is
$$ \begin{aligned} P(6 \leq X \leq 10)&=P(X\leq 10)-P(X\leq 6)\\ &=F(10) -F(6)\\ &=\bigg(1-\frac{1}{2}e^{\dfrac{-(10-5)}{2}}\bigg)-\bigg(1-\frac{1}{2}e^{\dfrac{-(6-5)}{2}}\bigg)\\ &=\frac{1}{2}e^{\dfrac{-(6-5)}{2}}-\frac{1}{2}e^{\dfrac{-(10-5)}{2}}\\ &= 0.3033-0.041\\ &=0.2623 \end{aligned} $$
d. The probability that $X$ is greater than $3.5$ is
$$ \begin{aligned} P(X > 3.5) &=1-P(X< 3.5)\\ &=1-F(3.5)\\ &=1-\frac{1}{2}e^{\dfrac{-(3.5-5)}{2}}\\ &\qquad (\because 3.5< mu)\\ &=1-\frac{1}{2}e^{\dfrac{(3.5-5)}{2}}\\ &= 1-0.2362\\ &= 0.7638 \end{aligned} $$
Conclusion
In this tutorial, you learned about how to calculate probabilities of Laplace distribution. You also learned about how to solve numerical problems based on Laplace distribution.
To read more about the step by step tutorial on Laplace distribution refer the link Laplace Distribution. This tutorial will help you to understand Laplace distribution and you will learn how to derive mean, median, quartiles of Laplace distribution, moment generating function and characteristics function of Laplace distribution and other properties of Laplace distribution.
To learn more about other discrete probability distributions, please refer to the following tutorial:
Let me know in the comments if you have any questions on Laplace Distribution Examples and your thought on this article.