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# Beta Scaled Distribution

<audio controls> <source src="../../_static/betascaled.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), [Bounded](../../gallery_tags.rst#bounded), [Symmetric](../../gallery_tags.rst#symmetric) (when α = β)

The Beta scaled distribution is a continuous probability distribution similar to the Beta distribution but instead of being bounded between 0 and1 it is bounded between $lower$ and $upper$. It is usually defined by two positive shape parameters: ($\alpha$) and ($\beta$). But other parametrization like mean ($\mu$) and concentration ($\nu$) are also common.

The Beta scaled distribution can adopt a wide range of "shapes" including uniform, U-shape, normal-like, exponential-like, and many others, always restricted to a given interval. This flexibility makes it a versatile choice for modeling random variables that are known to be bounded like percentages, grades, some physical quantities like temperature of liquid water at a given pressure. 

## Key properties and parameters

```{eval-rst}
========  ============================================================================
Support   :math:`x \in (lower, upper)`
Mean      :math:`\dfrac{\alpha}{\alpha + \beta} (upper-lower) + lower`
Variance  :math:`\dfrac{\alpha \beta}{(\alpha+\beta)^2(\alpha+\beta+1)} (upper-lower)`
========  ============================================================================
```

**Parameters:**

- $\alpha$ : (float) Shape parameter, $\alpha > 0$.
- $\beta$ : (float) Shape parameter, $\beta > 0$.
- $lower$ : (float) Lower bound of the distribution, $lower < upper$.
- $upper$ : (float) Upper bound of the distribution, $upper > lower$.

### Probability Density Function (PDF)

$$
f(x \mid \alpha, \beta, lower, upper) =
    \frac{(x-\text{lower})^{\alpha - 1} (\text{upper} - x)^{\beta - 1}}
    {(\text{upper}-\text{lower})^{\alpha+\beta-1} B(\alpha, \beta)}
$$

where $B(\alpha,\beta)$ is the [Beta function](https://en.wikipedia.org/wiki/Beta_function) 

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


from preliz import BetaScaled, style
style.use('preliz-doc')
alphas = [2, 2]
betas = [2, 5]
lowers = [-0.5, -1]
uppers = [1.5, 2]
for alpha, beta, lower, upper in zip(alphas, betas, lowers, uppers):
    BetaScaled(alpha, beta, lower, upper).plot_pdf()
```

### Cumulative Distribution Function (CDF)

$$
F(x \mid \alpha,\beta, lower, upper) = \frac{B(y;\alpha,\beta)}{B(\alpha,\beta)} = I_y(\alpha,\beta)
$$

where $y$ is the scaled variable $y = \frac{(x - lower)}{(upper - lower)}$. $B(x;\alpha,\beta)$ is the [Incomplete beta function](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function) and $I_x(\alpha,\beta)$ is the [regularized incomplete beta function](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function).

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

for alpha, beta, lower, upper in zip(alphas, betas, lowers, uppers):
    BetaScaled(alpha, beta, lower, upper).plot_cdf()
```

```{seealso}
:class: seealso

**Related Distributions:**
- [Beta](beta.md) - A Beta scaled distribution with $lower=0$ and $upper=1$.
- [Kumaraswamy](kumaraswamy.md) -  It is similar to the Beta scaled distribution, but restricted to the [0, 1] interval and with closed form expression for its probability density function, cumulative distribution function and quantile function.
- [Uniform](uniform.md) - The Uniform distribution on the interval $[lower, upper]$ is a special case of the Beta scaled distribution with $\alpha = \beta = 1$.
```

## References

- Wikipedia - [Beta distribution with four parameters](https://en.wikipedia.org/wiki/Beta_distribution#Four_parameters)