BETA

Overview

The BETA function provides a unified interface to the main methods of the Beta distribution, including PDF, CDF, inverse CDF, survival function, and distribution statistics.

Excel provides the BETA.DIST and BETA.INV functions, which can compute the PDF, CDF, and quantile (inverse CDF) for the beta distribution. The Python function in Excel here also supports the survival function, inverse survival, and distribution statistics (mean, median, variance, standard deviation), as well as location and scale parameters, which are not available in native Excel functions.

The Beta distribution is a continuous probability distribution defined on the interval [0, 1], parameterized by two positive shape parameters $a$ and $b$ . It is widely used in Bayesian statistics, modeling proportions, and random variables limited to intervals of finite length. The PDF is given by:

f(x, a, b) = \frac{\Gamma(a+b) x^{a-1} (1-x)^{b-1}}{\Gamma(a) \Gamma(b)}

for $0 \leq x \leq 1$ , $a > 0$ , $b > 0$ , where $\Gamma$ is the gamma function.

The Beta distribution is flexible and can take on a variety of shapes depending on the values of $a$ and $b$ . For more details, see the official SciPy documentation .

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=BETA(value, a, b, [loc], [scale], [method])

value (float, required for pdf, cdf, icdf, sf, isf):
- For pdf, cdf, sf: the value $x$ at which to evaluate the function (must be $0 \leq x \leq 1$ for standard beta)
- For icdf, isf: the probability $q$ (must be between 0 and 1)
- For mean, median, var, std: skip parameter
a (float, required): First shape parameter. Must be $> 0$ .
b (float, required): Second shape parameter. Must be $> 0$ .
loc (float, optional, default=0.0): Location parameter.
scale (float, optional, default=1.0): Scale parameter. Must be $> 0$ .
method (string, optional, default=“pdf”): One of pdf, cdf, icdf, sf, isf, mean, median, var, std.

Method	Description	Output
`pdf`	Probability Density Function: $f(x)$ , the likelihood of a specific value $x$ .	Density at $x$
`cdf`	Cumulative Distribution Function: $P(X \leq x)$ , the probability that $X$ is less than or equal to $x$ .	Probability
`icdf`	Inverse CDF (Quantile Function): Returns $x$ such that $P(X \leq x) = q$ for a given probability $q$ .	Value $x$
`sf`	Survival Function: $P(X > x)$ , the probability that $X$ is greater than $x$ .	Probability
`isf`	Inverse Survival Function: Returns $x$ such that $P(X > x) = q$ for a given probability $q$ .	Value $x$
`mean`	Mean (expected value) of the distribution.	Mean value
`median`	Median of the distribution.	Median value
`var`	Variance of the distribution.	Variance
`std`	Standard deviation of the distribution.	Standard deviation

The function returns a single value (float): the result of the requested method, or an error message (string) if the input is invalid.

Examples

Example 1: PDF at x=0.5

Inputs:

value	a	b	loc	scale	method
0.5	2	3	0	1	pdf

Excel formula:


=BETA(0.5, 2, 3, 0, 1, "pdf")

Expected output:

Result
1.5000000000000004

Example 2: CDF at x=0.5

Inputs:

value	a	b	loc	scale	method
0.5	2	3	0	1	cdf

Excel formula:


=BETA(0.5, 2, 3, 0, 1, "cdf")

Expected output:

Result
0.6875

Example 3: Inverse CDF (Quantile) at q=0.6875

Inputs:

value	a	b	loc	scale	method
0.6875	2	3	0	1	icdf

Excel formula:


=BETA(0.6875, 2, 3, 0, 1, "icdf")

Expected output:

Result
0.5

Example 4: Mean of the distribution

Inputs:

value	a	b	loc	scale	method
	2	3	0	1	mean

Excel formula:


=BETA( , 2, 3, 0, 1, "mean")

Expected output:

Result
0.4

Python Code


from scipy.stats import beta as scipy_beta
import math
 
def beta(value=None, a=1.0, b=1.0, loc=0.0, scale=1.0, method="pdf"):
    """
    Generalized Beta distribution function supporting multiple methods.
 
    Args:
        value: Input value (float), required for methods except 'mean', 'median', 'var', 'std'.
        a: First shape parameter (float, >0).
        b: Second shape parameter (float, >0).
        loc: Location parameter (float, default: 0.0).
        scale: Scale parameter (float, default: 1.0, >0).
        method: Which method to compute (str): 'pdf', 'cdf', 'icdf', 'sf', 'isf', 'mean', 'median', 'var', 'std'. Default is 'pdf'.
 
    Returns:
        Result of the requested method (float or str), or an error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    valid_methods = ['pdf', 'cdf', 'icdf', 'sf', 'isf', 'mean', 'median', 'var', 'std']
    if not isinstance(method, str) or method.lower() not in valid_methods:
        return f"Invalid method: {method}. Must be one of {valid_methods}."
    method = method.lower()
    try:
        a = float(a)
        b = float(b)
        loc = float(loc)
        scale = float(scale)
    except Exception:
        return "Invalid input: a, b, loc, and scale must be numbers."
    if a <= 0:
        return "Invalid input: a must be > 0."
    if b <= 0:
        return "Invalid input: b must be > 0."
    if scale <= 0:
        return "Invalid input: scale must be > 0."
    dist = scipy_beta(a, b, loc, scale)
    # Methods that require value
    if method in ['pdf', 'cdf', 'icdf', 'sf', 'isf']:
        if value is None:
            return f"Invalid input: missing required argument 'value' for method '{method}'."
        try:
            value = float(value)
        except Exception:
            return "Invalid input: value must be a number."
        if method in ['pdf', 'cdf', 'sf'] and not (loc <= value <= loc + scale):
            return "Invalid input: value must be within [loc, loc + scale] for this method."
        if method in ['isf', 'icdf'] and not (0 <= value <= 1):
            return "Invalid input: value (probability) must be between 0 and 1 for isf/icdf."
        try:
            if method == 'pdf':
                result = dist.pdf(value)
            elif method == 'cdf':
                result = dist.cdf(value)
            elif method == 'sf':
                result = dist.sf(value)
            elif method == 'isf':
                result = dist.isf(value)
            elif method == 'icdf':
                result = dist.ppf(value)
        except Exception as e:
            return f"scipy.stats.beta error: {e}"
        if isinstance(result, float):
            if math.isnan(result):
                return "Result is NaN (not a number)"
            if math.isinf(result):
                return "inf" if result > 0 else "-inf"
        return result
    # Methods that do not require value
    try:
        if method == 'mean':
            result = dist.mean()
        elif method == 'median':
            result = dist.median()
        elif method == 'var':
            result = dist.var()
        elif method == 'std':
            result = dist.std()
    except Exception as e:
        return f"scipy.stats.beta error: {e}"
    if isinstance(result, float):
        if math.isnan(result):
            return "Result is NaN (not a number)"
        if math.isinf(result):
            return "inf" if result > 0 else "-inf"
    return result

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BETA

Overview

Usage

Examples

Python Code

Live Notebook

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