BOLTZMANN

Overview

The BOLTZMANN function computes values related to the Boltzmann (Truncated Discrete Exponential) distribution, a discrete probability distribution with support on {0, ..., N-1}. 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 Boltzmann distribution function. The Python function in Excel provided here supports PMF, CDF, SF, ICDF, ISF, and distribution statistics (mean, median, variance, standard deviation), which are not available natively in Excel.

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

Usage

To use the function in Excel:


=BOLTZMANN(k, lambda, N, [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 value 0 <= k < N. For statistics modes, this is ignored and can be set to 0.
lambda (float, required): Rate parameter lambda > 0.
N (int, required): Number of possible values N > 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=3, lambda=1.4, N=19

Inputs:

k	lambda	N	mode	loc
3	1.4	19	pmf	0

Excel formula:


=BOLTZMANN(3, 1.4, 19, "pmf", 0)

Expected output:

Result
0.0113

Example 2: CDF at k=3, lambda=1.4, N=19

Inputs:

k	lambda	N	mode	loc
3	1.4	19	cdf	0

Excel formula:


=BOLTZMANN(3, 1.4, 19, "cdf", 0)

Expected output:

Result
0.9963

Example 3: Survival Function at k=3, lambda=1.4, N=19

Inputs:

k	lambda	N	mode	loc
3	1.4	19	sf	0

Excel formula:


=BOLTZMANN(3, 1.4, 19, "sf", 0)

Expected output:

Result
0.0037

Example 4: Inverse CDF (ICDF) for probability k=0.5, lambda=1.4, N=19

Inputs:

k	lambda	N	mode	loc
0.5	1.4	19	icdf	0

Excel formula:


=BOLTZMANN(0.5, 1.4, 19, "icdf", 0)

Expected output:

Result
0

Example 5: Mean, Variance, Std, Median

Inputs:

lambda	N	mode
1.4	19	mean
1.4	19	var
1.4	19	std
1.4	19	median

Excel formulas:


=BOLTZMANN(0, 1.4, 19, "mean", 0)
=BOLTZMANN(0, 1.4, 19, "var", 0)
=BOLTZMANN(0, 1.4, 19, "std", 0)
=BOLTZMANN(0, 1.4, 19, "median", 0)

Expected outputs:

Result
0.3273
0.4344
0.6591
0

Python Code


from scipy.stats import boltzmann as scipy_boltzmann
 
def boltzmann(k, lambda_, N, mode="pmf", loc=0):
    """
    Compute Boltzmann distribution values: PMF, CDF, SF, ICDF, ISF, mean, variance, std, or median.
 
    Args:
        k: Value(s) at which to evaluate (float or 2D list).
        lambda_: Rate parameter (float, > 0).
        N: Number of possible values (int, > 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 lambda_
    try:
        lam_val = float(lambda_)
        if not (lam_val > 0):
            return "Invalid input: lambda must be > 0."
    except Exception:
        return "Invalid input: lambda must be a number."
    # Validate N
    try:
        N_val = int(N)
        if not (N_val > 0):
            return "Invalid input: N must be > 0."
    except Exception:
        return "Invalid input: N must be an integer."
    # 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_boltzmann.mean(lam_val, N_val, loc=loc_val)
    if mode == "var":
        return scipy_boltzmann.var(lam_val, N_val, loc=loc_val)
    if mode == "std":
        return scipy_boltzmann.std(lam_val, N_val, loc=loc_val)
    if mode == "median":
        return scipy_boltzmann.median(lam_val, N_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_boltzmann.pmf(kval, lam_val, N_val, loc=loc_val))
        elif mode == "cdf":
            return float(scipy_boltzmann.cdf(kval, lam_val, N_val, loc=loc_val))
        elif mode == "sf":
            return float(scipy_boltzmann.sf(kval, lam_val, N_val, loc=loc_val))
        elif mode == "icdf":
            return float(scipy_boltzmann.ppf(kval, lam_val, N_val, loc=loc_val))
        elif mode == "isf":
            return float(scipy_boltzmann.isf(kval, lam_val, N_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