Cauchy Distribution Calculator

Location (

μ

Scale (

γ

Probability Type:

X value (

x_{1}

Upper bound (

x_{2}

Output Type: PDF Chart CDF Chart Data Table

Decimal Places: 2 3 5 8

Introduction

The Cauchy distribution is a continuous probability distribution characterised by a location parameter $μ$ and a scale parameter $γ$ . These parameters determine the centre and spread of the distribution, which is notable for its heavy tails and the absence of finite moments such as the mean and variance. Evaluating the distribution for specific values of the random variable $X$ allows probabilities and density behaviour to be examined within a formal statistical framework.

What this calculator does

The tool performs probability density and cumulative probability calculations for the Cauchy distribution. Users provide the location parameter, scale parameter, and specific $x$ values to determine the likelihood of observations being less than, greater than, or between given bounds. It produces numerical probabilities, step-by-step calculation processes, and visual representations including probability density function (PDF) and cumulative distribution function (CDF) charts or data tables.

Formula used

The probability density function (PDF) calculates the density at a point $x$ using the location $μ$ and scale $γ$ . The cumulative distribution function (CDF) determines the area under the curve from negative infinity to $x$ using the arctangent function.

f (x| μ| γ) = \frac{1}{π γ [1 + {(\frac{x - μ}{γ})}^{2}]}

F (x| μ| γ) = \frac{1}{π} {tan}^{-1} (\frac{x - μ}{γ}) + 0.5

How to use this calculator

1. Enter the location $μ$ and positive scale $γ$ parameters.
2. Select the probability type: less than, greater than, or between bounds.
3. Input the specific $x$ value or values for the calculation.
4. Choose the desired output format and execute the calculation to view results.

Example calculation

Scenario: A student in a physics lab is modelling the distribution of light intensity on a screen from a point source to determine the probability of a hit.

Inputs: Location $μ$ = $0$ , Scale $γ$ = $1$ , and $X$ value $x_{1}$ = $1$ for $P (X \leq 1)$ .

Working:

Step 1: $z_{1} = \frac{x_{1} - μ}{γ}$

Step 2: $\frac{1 - 0}{1} = 1$

Step 3: $\frac{1}{π} {tan}^{-1} (1) + 0.5$

Step 4: $0.25 + 0.5 = 0.75$

Result: 0.75

Interpretation: There is a 75% probability that an observed value will be less than or equal to 1 in this standard distribution.

Summary: The calculation successfully determines the cumulative probability for the specified point.

Understanding the result

The resulting probability indicates the likelihood of an outcome occurring within the specified range. A PDF value represents the relative likelihood at a point, while the CDF value provides the area under the curve, showing the total probability from the lower limit to the chosen $x$ value.

Assumptions and limitations

This model assumes the data follows a Cauchy distribution, meaning it possesses heavy tails and no defined mean or variance. The scale parameter must always be a positive value greater than zero to ensure a valid probability density.

Common mistakes to avoid

A frequent error is attempting to calculate a mean or standard deviation for the Cauchy distribution, as these moments do not converge. Users should also ensure that the scale parameter is not set to zero and that the upper bound is strictly greater than the lower bound.

Sensitivity and robustness

The Cauchy distribution is highly sensitive to the scale parameter, which dictates the "fatness" of the tails. Small changes in location shift the entire curve. Because of its heavy tails, the distribution is robust against outliers in the sense that extreme values are more expected than in a normal distribution.

Troubleshooting

If the results appear incorrect, verify that the scale parameter is positive and that inputs are numeric. Scientific notation and excessive decimal places are not supported. Ensure the CSRF token is valid by refreshing the page if a session error occurs during the calculation process.

Frequently asked questions

Why does the Cauchy distribution have no mean?

The tails of the distribution are so heavy that the integral for the expected value does not converge, making the mean undefined in a mathematical sense.

What is the role of the location parameter?

The location parameter specifies the peak of the distribution and represents its median and mode, indicating where the distribution is centred on the x-axis.

How does the scale parameter affect the chart?

A larger scale parameter widens the distribution and lowers the peak of the density function, while a smaller scale results in a narrower, sharper peak.

Where this calculation is used

In academic settings, the Cauchy distribution is a staple of probability theory, often used as a counterexample to the Central Limit Theorem because it does not have a finite variance. It appears in physics to describe Breit-Wigner resonance and in social research to model variables with extreme outliers that occur more frequently than predicted by a normal distribution. Students use this calculator to visualise how heavy tails influence cumulative probabilities compared to more common bell curves studied in introductory statistics.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.