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# Chi-Squared Distribution

<audio controls> <source src="../../_static/chisquared.mp3" type="audio/mpeg"> This browser cannot play the pronunciation audio file for this distribution. </audio>

[Univariate](../../gallery_tags.rst#univariate), [Continuous](../../gallery_tags.rst#continuous), [Non-Negative](../../gallery_tags.rst#non-negative), [Light-tailed](../../gallery_tags.rst#light-tailed)

The chi-squared (also chi-square or $\chi^2$) is a continuous probability distribution characterized by a single parameter, $\nu$, usually called degrees of freedom. This distribution emerges from the sum of the squares of $\nu$ independent standard normal random variables.

The chi-squared distribution is widely used in many statistical tests, for hypothesis testing and constructing confidence intervals. 

## Key properties and parameters

```{eval-rst}
========  ==========================================
Support   :math:`x \in [0, \infty)`
Mean      :math:`\nu`
Variance  :math:`2\nu`
========  ==========================================
```

**Parameters:**

- $\nu$ : (float) Degrees of freedom, $\nu > 0$.

### Probability Density Function (PDF)

$$
f(x|\nu) = \frac{1}{2^{\nu/2}\Gamma(\nu/2)} x^{\nu/2 - 1} e^{-x/2}
$$

where $\Gamma$ is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function).

```{code-cell}
---
tags: [remove-input]
mystnb:
  image:
    alt: Chi-Squared Distribution PDF
---

import numpy as np

from preliz import ChiSquared, style
style.use('preliz-doc')
nus = [1, 3, 9]

for nu in nus:
    ax = ChiSquared(nu).plot_pdf(support=(np.finfo(float).eps, 20))
    ax.set_ylim(0, 0.6)
```

### Cumulative Distribution Function (CDF)

$$
F(x|\nu) = \frac{1}{\Gamma(\nu/2)} \gamma(\nu/2, x/2)
$$

where $\gamma$ is the [lower incomplete gamma function](https://en.wikipedia.org/wiki/Incomplete_gamma_function).

```{code-cell}
---
tags: [remove-input]
mystnb:
  image:
    alt: Chi-Squared Distribution CDF
---

for nu in nus:
    ChiSquared(nu).plot_cdf(support=(0, 20))
```

```{seealso}
:class: seealso

**Common Alternatives:**

- [Gamma](gamma.md) - The chi-squared distribution is a special case of the gamma distribution with the shape parameter $\alpha = \nu/2$ and the scale parameter $\beta = 1/2$.

**Related Distributions:**

- [Normal](normal.md) - By definition, the chi-squared distribution is the sum of the squares of $\nu$ independent standard normal random variables.
- [Exponential](exponential.md) - A chi-squared distribution with 2 degrees of freedom is equivalent to an exponential distribution with the rate parameter $\lambda = 1/2$, because the exponential distribution is also a special case of the gamma distribution.
```

## References

- [Wikipedia - Chi-squared distribution](https://en.wikipedia.org/wiki/Chi-squared_distribution)