Cauchy Distribution Calculator With Examples

Cauchy Distribution Calculator

Use this calculator to find the probability density and cumulative probabilities for Cauchy distribution with parameter $a$ and $b$.

How to find Cauchy Distribution Probabilities?

Step 1 - Enter the location parameter $\mu$

Step 2 - Enter the scale parameter $\lambda$

Step 2 - Enter the value of $x$

Step 4 - Click on "Calculate" button to get Cauchy distribution probabilities

Step 5 - Gives the output probability at $x$ for Cauchy distribution

Step 6 - Gives the output cumulative probabilities for Cauchy distribution

Definition Cauchy Distribution

A continuous random variable $X$ is said to follow Cauchy distribution with parameters $\mu$ and $\lambda$ if its probability density function is given by

$$ \begin{align*} f(x; \mu, \lambda) &= \begin{cases} \frac{\lambda}{\pi}\cdot \frac{1}{\lambda^2+(x-\mu)^2}, & -\infty < x < \infty; \\ & -\infty < \mu < \infty, \lambda > 0; \\ 0, & Otherwise. \end{cases} \end{align*} $$

Distribution Function of Cauchy Distribution

The distribution function of Cauchy distribution is

$$ \begin{equation*} F(x) =\frac{1}{\pi}\tan^{-1}\bigg(\frac{x-\mu}{\lambda}\bigg) + \frac{1}{2}. \end{equation*} $$

Cauchy Distribution Example

Let $X\sim C(2,4)$. Find the probability that

a. $X$ is less than 3,

b. $X$ is greater than 4,

c. $X$ is between 1 and 3.5

Solution

The distribution function of Cauchy distribution is

$$ \begin{aligned} F(x) &= \frac{1}{2}+ \frac{1}{\pi}tan^{-1}\big(\frac{x-\mu}{\lambda}\big)\\ &=0.5+\frac{1}{\pi} tan^{-1}\big(\frac{x-2}{4}\big) \end{aligned} $$

a. The probability that $X$ is less than $3$ is

$$ \begin{aligned} P(X \leq 3) &=F(3)\\ &=0.5+\frac{1}{\pi} tan^{-1}\big(\frac{3-2}{4}\big)\\ &=0.5 + \frac{1}{3.1416}tan^{-1}\big(0.25\big)\\ &=0.5 + \frac{1}{3.1416}(0.245)\\ &= 0.578 \end{aligned} $$

b. The probability that $X$ is greater than $4$ is

$$ \begin{aligned} P(X > 4) &=1- P(X < 4)\\ &= 1- F(4)\\ &=1-\bigg(0.5+\frac{1}{\pi} tan^{-1}\big(\frac{4-2}{4}\big)\bigg)\\ &=0.5 - \frac{1}{3.1416}tan^{-1}\big(0.5\big)\\ &=0.5 - \frac{1}{3.1416}(0.4636)\\ &= 0.3524 \end{aligned} $$

c. The probability that $X$ is between $1$ and $3$ is

$$ \begin{aligned} P(1 \leq X \leq 3)&=P(X\leq 3)-P(X\leq 1)\\ &=F(3) -F(1)\\ &=\bigg[0.5+\frac{1}{\pi} tan^{-1}\big(\frac{3-2}{4}\big)\bigg]-\bigg[0.5+\frac{1}{\pi} tan^{-1}\big(\frac{1-2}{4}\big)\bigg]\\ &=\frac{1}{\pi} tan^{-1}\big(0.25\big)-\frac{1}{\pi} tan^{-1}\big(-0.25\big)\\ &=\frac{1}{3.1416}(0.245)-\frac{1}{3.1416}(-0.245)\\ &=0.156 \end{aligned} $$

Conclusion

In this tutorial, you learned about how to calculate median, quartiles and probabilities of Cauchy distribution. You also learned about how to solve numerical problems based on Cauchy distribution.

To read more about the step by step tutorial on Cauchy distribution refer the link Cauchy Distribution. This tutorial will help you to understand Cauchy distribution and you will learn how to derive median of Cauchy distribution, mode of Cauchy distribution, characteristics function and other properties of Cauchy distribution.

To learn more about other discrete probability distributions, please refer to the following tutorial:

Probability distributions

Let me know in the comments if you have any questions on Cauchy Distribution Examples and your thought on this article.