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 $b$.

Cauchy 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 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.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.

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