BETANBINOM

Overview

The BETANBINOM function computes values related to the Beta-negative-binomial discrete distribution, which describes the number of failures before n successes in a sequence of Bernoulli trials where the probability of success follows a beta distribution. This function can return the probability mass function (PMF), cumulative distribution function (CDF), survival function (SF), inverse CDF (quantile/ICDF), inverse SF (ISF), mean, variance, standard deviation, or median for a given value.

Excel does not provide a native Beta-negative-binomial distribution function. The Python function in Excel provided here supports all major distribution features, including PMF, CDF, survival function, inverse CDF (quantile), inverse survival function, and distribution statistics (mean, median, variance, standard deviation).

For more details, see the scipy.stats.betanbinom documentation .

Usage

To use the function in Excel:


=BETANBINOM(k, n, a, b, [mode], [loc])

k (float or 2D list, required): Value(s) at which to evaluate the distribution. For PMF, CDF, SF, ICDF, and ISF, this is the integer value (k >= 0).
n (int, required): Number of successes (n >= 0).
a (float, required): Alpha parameter of the beta distribution (a > 0).
b (float, required): Beta parameter of the beta distribution (b > 0).
mode (str, optional, default=“pmf”): Output type. One of "pmf", "cdf", "sf", "icdf", "isf", "mean", "var", "std", or "median".
loc (float, optional, default=0): Location parameter (shifts the distribution).

The function returns a scalar or 2D list of floats (for array input), or an error message (string) if the input is invalid. The output depends on the selected mode:

pmf: Probability mass function at k.
cdf: Cumulative distribution function at k.
sf: Survival function (1 - CDF) at k.
icdf: Inverse CDF (quantile) for probability k.
isf: Inverse survival function for probability k.
mean: Mean of the distribution.
var: Variance of the distribution.
std: Standard deviation of the distribution.
median: Median of the distribution.

Examples

Example 1: PMF at k=2, n=5, a=9.3, b=1

Inputs:

k	n	a	b	mode	loc
2	5	9.3	1	pmf	0

Excel formula:


=BETANBINOM(2, 5, 9.3, 1, "pmf", 0)

Expected output:

Result
0.0782

Example 2: CDF at k=2, n=5, a=9.3, b=1

Inputs:

k	n	a	b	mode	loc
2	5	9.3	1	cdf	0

Excel formula:


=BETANBINOM(2, 5, 9.3, 1, "cdf", 0)

Expected output:

Result
0.9411

Example 3: Survival Function at k=2, n=5, a=9.3, b=1

Inputs:

k	n	a	b	mode	loc
2	5	9.3	1	sf	0

Excel formula:


=BETANBINOM(2, 5, 9.3, 1, "sf", 0)

Expected output:

Result
0.0589

Example 4: Inverse CDF (ICDF) for probability k=0.5, n=5, a=9.3, b=1

Inputs:

k	n	a	b	mode	loc
0.5	5	9.3	1	icdf	0

Excel formula:


=BETANBINOM(0.5, 5, 9.3, 1, "icdf", 0)

Expected output:

Result
0

Example 5: Mean, Variance, Std, Median

Inputs:

n	a	b	mode
5	9.3	1	mean
5	9.3	1	var
5	9.3	1	std
5	9.3	1	median

Excel formulas:


=BETANBINOM(0, 5, 9.3, 1, "mean", 0)
=BETANBINOM(0, 5, 9.3, 1, "var", 0)
=BETANBINOM(0, 5, 9.3, 1, "std", 0)
=BETANBINOM(0, 5, 9.3, 1, "median", 0)

Expected outputs:

Result
0.6024
1.2298
1.1090
0

Python Code


from scipy.stats import betanbinom as scipy_betanbinom
 
def betanbinom(k, n, a, b, mode="pmf", loc=0):
    """
    Compute Beta-negative-binomial distribution values: PMF, CDF, SF, ICDF, ISF, mean, variance, std, or median.
 
    Args:
        k: Value(s) at which to evaluate (float or 2D list).
        n: Number of successes (int, n >= 0).
        a: Alpha parameter (float, a > 0).
        b: Beta parameter (float, b > 0).
        mode: Output type: 'pmf', 'cdf', 'sf', 'icdf', 'isf', 'mean', 'var', 'std', or 'median'.
        loc: Location parameter (float, default 0).
 
    Returns:
        Scalar or 2D list of floats, or error message (str) if invalid.
    """
    # Validate n, a, b
    try:
        n_val = int(n)
        a_val = float(a)
        b_val = float(b)
        if not (n_val >= 0 and a_val > 0 and b_val > 0):
            return "Invalid input: n must be >= 0, a and b must be > 0."
    except Exception:
        return "Invalid input: n must be integer, a and b must be numbers."
    # Validate loc
    try:
        loc_val = float(loc)
    except Exception:
        return "Invalid input: loc must be a number."
    # Validate mode
    valid_modes = ["pmf", "cdf", "sf", "icdf", "isf", "mean", "var", "std", "median"]
    if not isinstance(mode, str) or mode not in valid_modes:
        return f"Invalid input: mode must be one of {valid_modes}."
    # Helper to process k (scalar or 2D list)
    def process_k(val):
        try:
            return float(val)
        except Exception:
            return None
    # Handle statistics
    if mode == "mean":
        return scipy_betanbinom.mean(n_val, a_val, b_val, loc=loc_val)
    if mode == "var":
        return scipy_betanbinom.var(n_val, a_val, b_val, loc=loc_val)
    if mode == "std":
        return scipy_betanbinom.std(n_val, a_val, b_val, loc=loc_val)
    if mode == "median":
        return scipy_betanbinom.median(n_val, a_val, b_val, loc=loc_val)
    # PMF, CDF, SF, ICDF, ISF
    def compute(val):
        kval = process_k(val)
        if kval is None:
            return "Invalid input: k must be a number."
        if mode == "pmf":
            return float(scipy_betanbinom.pmf(kval, n_val, a_val, b_val, loc=loc_val))
        elif mode == "cdf":
            return float(scipy_betanbinom.cdf(kval, n_val, a_val, b_val, loc=loc_val))
        elif mode == "sf":
            return float(scipy_betanbinom.sf(kval, n_val, a_val, b_val, loc=loc_val))
        elif mode == "icdf":
            return float(scipy_betanbinom.ppf(kval, n_val, a_val, b_val, loc=loc_val))
        elif mode == "isf":
            return float(scipy_betanbinom.isf(kval, n_val, a_val, b_val, loc=loc_val))
    # 2D list or scalar
    if isinstance(k, list):
        # 2D list
        if not all(isinstance(row, list) for row in k):
            return "Invalid input: k must be a scalar or 2D list."
        result = []
        for row in k:
            result_row = []
            for val in row:
                out = compute(val)
                if isinstance(out, str):
                    return out
                result_row.append(out)
            result.append(result_row)
        return result
    else:
        return compute(k)

Live Demo

Example Workbook

Link to Workbook