Probability Distributions Cheat Sheet

Reference for the most common discrete and continuous probability distributions, their parameters, and how to work with them in Python using scipy.stats.

2 PagesBeginnerMar 5, 2026

Working with Distributions

PDF, CDF, PMF, and sampling with scipy.stats.

python

from scipy import stats# Normal (Gaussian) distributionnorm = stats.norm(loc=0, scale=1)          # mean=0, std=1print(norm.pdf(0))                          # probability density at x=0print(norm.cdf(1.96))                       # P(X <= 1.96)samples = norm.rvs(size=1000)               # random samples# Binomial distributionbinom = stats.binom(n=10, p=0.5)            # 10 trials, p=0.5 successprint(binom.pmf(5))                         # P(exactly 5 successes)# Poisson distributionpoisson = stats.poisson(mu=3)               # average rate = 3 eventsprint(poisson.pmf(2))                       # P(exactly 2 events)# Uniform distributionuniform = stats.uniform(loc=0, scale=10)    # range [0, 10]

Fitting & Goodness-of-Fit

Fit a distribution to data and test normality.

python

from scipy import statsdata = [23, 25, 21, 22, 24, 20, 26, 27, 19, 24]# Fit a normal distribution to data (maximum likelihood estimate)mu, sigma = stats.norm.fit(data)# Test if data plausibly comes from a normal distributionstat, p_value = stats.shapiro(data)   # Shapiro-Wilk normality testprint(f"p = {p_value:.4f}")           # p < 0.05 suggests non-normal# Kolmogorov-Smirnov test against a specific fitted distributionks_stat, p_value = stats.kstest(data, "norm", args=(mu, sigma))

Distribution Reference

When to use each common distribution.

Normal (Gaussian)- continuous, symmetric bell curve; parameters mean μ and std σ; models measurement errors, heights
Binomial- discrete count of successes in n independent trials with success probability p
Bernoulli- special case of binomial with n=1; a single yes/no trial
Poisson- discrete count of events in a fixed interval given average rate λ; models rare, independent events
Exponential- continuous time between events in a Poisson process; has the memoryless property
Uniform- all outcomes in a range are equally likely
Chi-square- distribution of the sum of squared standard normals; used in hypothesis tests
Student's t- like normal but heavier tails; used for small-sample inference with unknown population std

Key Properties

Statistics used to summarize any distribution.

Mean (expected value)- the long-run average outcome of the distribution
Variance / standard deviation- measures spread around the mean
Skewness- measures asymmetry; positive skew has a long right tail
Kurtosis- measures tail heaviness relative to a normal distribution
PDF vs PMF- PDF describes continuous distributions (density), PMF describes discrete ones (exact probability)
CDF- cumulative distribution function; gives P(X <= x) for any x

Pro Tip

The Central Limit Theorem means the sampling distribution of a mean approaches normal as sample size grows, regardless of the underlying distribution — this is why t-tests and z-tests work reasonably well even on non-normal data once n is large enough (roughly n ≥ 30).

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