Probability Distributions

A probability distribution assigns probabilities to outcomes of a random variable; key distributions model common data-generating processes.

Ask the Data science assistant1 min read · Updated September 9, 2026

Definition

Binomial $B(n, p)$ : number of successes in $n$ independent Bernoulli trials, PMF $P(k) = \binom{n}{k}p^k(1-p)^{n-k}$ .

Normal (Gaussian) $N(\mu, \sigma^2)$ : PDF $\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}$ ; symmetric, bell-shaped, defined by mean $\mu$ and std $\sigma$ .

Intuition

The central limit theorem says the sum/mean of i.i.d. random variables converges to normal as $n$ grows — this is why normal appears everywhere.

Poisson models "events per interval" — arrivals per minute, defects per item — with a single rate parameter $\lambda$ .

Worked example

Heights of adult humans approximate $N(170, 9^2)$ cm; IQ is centered at 100 with std 15.

Number of spam emails in an hour might follow Poisson(7) if you average 7 per hour.

The math

Normal properties: 68% of mass within $1\sigma$ , 95% within $2\sigma$ , 99.7% within $3\sigma$ of $\mu$ .

Standard normal $Z = (X - \mu)/\sigma$ ; used to compute normal probabilities via $Z$ -tables or scipy.stats.norm.cdf.

In practice

Assume normal errors in linear regression; check residuals visually to validate.

Use Poisson regression for count data; binomial for binary outcomes.

More in Data science

Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.