Discrete Distributions

Overview

Discrete distributions describe variables that can assume only a countable number of values, such as integers (0, 1, 2…). Examples include coin flips, dice rolls, or the number of customers arriving at a store.

Common Distributions

BINOM: Binomial distribution. Number of successes in n independent Bernoulli trials (e.g., number of heads in 10 flips).
POISSON_DIST: Poisson distribution. Number of events occurring in a fixed interval of time or space (e.g., calls per hour).
GEOM: Geometric distribution. Number of trials needed to get the first success.
HYPERGEOM: Hypergeometric distribution. Sampling without replacement (e.g., drawing 5 cards from a deck).
NBINOM: Negative Binomial distribution. Number of failures before achieving n successes. Also used for over-dispersed count data.

<>:16: SyntaxWarning: invalid escape sequence '\l'
<>:16: SyntaxWarning: invalid escape sequence '\l'
C:\Users\brent\AppData\Local\Temp\ipykernel_11452\613262124.py:16: SyntaxWarning: invalid escape sequence '\l'
  ax.bar(x + width/2, pmf_pois, width, label=f'Poisson ($\lambda$={lambda_val})', color='orange', alpha=0.7)

Figure 1: Binomial vs Poisson: For large n and small p (like n=20, p=0.15), the Binomial distribution (teal) is well-approximated by the Poisson distribution (orange) with lambda = n*p = 3.

Specialized Distributions

ZIPF: Zipf’s Law. Frequency of words in a corpus, or city populations.
BOLTZMANN: Used in statistical mechanics for the probability of energy states.
SKELLAM: Distribution of the difference between two independent Poisson variables (e.g., difference in scores between two soccer teams).

Native Excel Capabilities

Excel has simpler discrete functions: BINOM.DIST, POISSON.DIST, HYPGEOM.DIST. However, it lacks coverage for simpler variants like Bernoulli (often just treated as Binomial n=1) or complex ones like Skellam, Zipf, or Plancke.

Tools

Tool	Description
BERNOULLI	Calculates properties of a Bernoulli discrete random variable.
BETABINOM	Compute Beta-binomial distribution values from scipy.stats.betabinom.
BETANBINOM	Compute Beta-negative-binomial distribution values: PMF, CDF, SF, ICDF, ISF, mean, variance, std, or median.
BINOM	Compute Binomial distribution values: PMF, CDF, SF, ICDF, ISF, mean, variance, std, or median.
BOLTZMANN	Compute Boltzmann distribution values: PMF, CDF, SF, ICDF, ISF, mean, variance, std, or median.
DLAPLACE	Compute Discrete Laplace distribution values: PMF, CDF, SF, ICDF, ISF, mean, variance, std, or median.
GEOM	Compute Geometric distribution values using scipy.stats.geom.
HYPERGEOM	Compute Hypergeometric distribution values: PMF, CDF, SF, ICDF, ISF, mean, variance, std, or median.
LOGSER	Compute Log-Series distribution values: PMF, CDF, SF, ICDF, ISF, mean, variance, std, or median.
NBINOM	Compute Negative Binomial distribution values using scipy.stats.nbinom.
NHYPERGEOM	Compute Negative Hypergeometric distribution values using scipy.stats.nhypergeom.
PLANCK	Compute Planck distribution values using scipy.stats.planck.
POISSON_DIST	Compute Poisson distribution values using scipy.stats.poisson.
RANDINT	Compute Uniform discrete distribution values: PMF, CDF, SF, ICDF, ISF, mean, variance, std, or median.
SKELLAM	Compute Skellam distribution values using scipy.stats.skellam.
YULESIMON	Compute Yule-Simon distribution values using scipy.stats.yulesimon.
ZIPF	Compute Zipf distribution values: PMF, CDF, SF, ICDF, ISF, mean, variance, std, or median.
ZIPFIAN	Compute Zipfian distribution values: PMF, CDF, SF, ICDF, ISF, mean, variance, std, or median.