NBINOM

Overview

The NBINOM function computes values related to the Negative Binomial distribution, a discrete probability distribution that describes the number of failures before a fixed number of successes occurs in independent Bernoulli trials, each with the same probability of success. 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.

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

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

Usage

To use the function in Excel:


=NBINOM(k, n, p, [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 event count (number of failures). For statistics modes, this is ignored and can be set to 0.
n (int, required): Number of successes (must be >= 0).
p (float, required): Probability of success (0 < p <= 1).
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=3, n=5, p=0.5

Inputs:

k	n	p	mode	loc
3	5	0.5	pmf	0

Excel formula:


=NBINOM(3, 5, 0.5, "pmf", 0)

Expected output:

Result
0.1367

Example 2: CDF at k=3, n=5, p=0.5

Inputs:

k	n	p	mode	loc
3	5	0.5	cdf	0

Excel formula:


=NBINOM(3, 5, 0.5, "cdf", 0)

Expected output:

Result
0.3633

Example 3: Survival Function at k=3, n=5, p=0.5

Inputs:

k	n	p	mode	loc
3	5	0.5	sf	0

Excel formula:


=NBINOM(3, 5, 0.5, "sf", 0)

Expected output:

Result
0.6367

Example 4: Inverse CDF (ICDF) for probability k=0.5, n=5, p=0.5

Inputs:

k	n	p	mode	loc
0.5	5	0.5	icdf	0

Excel formula:


=NBINOM(0.5, 5, 0.5, "icdf", 0)

Expected output:

Result
4

Example 5: Mean, Variance, Std, Median

Inputs:

n	p	mode
5	0.5	mean
5	0.5	var
5	0.5	std
5	0.5	median

Excel formulas:


=NBINOM(0, 5, 0.5, "mean", 0)
=NBINOM(0, 5, 0.5, "var", 0)
=NBINOM(0, 5, 0.5, "std", 0)
=NBINOM(0, 5, 0.5, "median", 0)

Expected outputs:

Result
5
10
3.1623
4

Python Code


from scipy.stats import nbinom as scipy_nbinom
 
def nbinom(k, n, p, mode="pmf", loc=0):
    """
    Compute 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, >=0).
        p: Probability of success (float, 0 < p <= 1).
        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
    try:
        n_val = int(n)
        if n_val < 0:
            return "Invalid input: n must be >= 0."
    except Exception:
        return "Invalid input: n must be an integer."
    # Validate p
    try:
        p_val = float(p)
        if not (0 < p_val <= 1):
            return "Invalid input: p must be between 0 (exclusive) and 1 (inclusive)."
    except Exception:
        return "Invalid input: p must be a number."
    # 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_nbinom.mean(n_val, p_val, loc=loc_val)
    if mode == "var":
        return scipy_nbinom.var(n_val, p_val, loc=loc_val)
    if mode == "std":
        return scipy_nbinom.std(n_val, p_val, loc=loc_val)
    if mode == "median":
        return scipy_nbinom.median(n_val, p_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_nbinom.pmf(kval, n_val, p_val, loc=loc_val))
        elif mode == "cdf":
            return float(scipy_nbinom.cdf(kval, n_val, p_val, loc=loc_val))
        elif mode == "sf":
            return float(scipy_nbinom.sf(kval, n_val, p_val, loc=loc_val))
        elif mode == "icdf":
            return float(scipy_nbinom.ppf(kval, n_val, p_val, loc=loc_val))
        elif mode == "isf":
            return float(scipy_nbinom.isf(kval, n_val, p_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