Probability Distributions
Normal, binomial and Poisson probabilities: P(X = x), P(X ≤ x), P(X ≥ x), the probability between two values, the normal quantile, mean, variance and a table.
Density or mass at x, lower and upper tail probabilities, the probability between two bounds, a quantile (normal), mean, variance and standard deviation, and a table around x.
Example: Normal μ 100, σ 15 at 130: z = 2, P(X ≥ 130) = 2.28 %, P(85 ≤ X ≤ 130) = 81.9 %, 97.5 % quantile 129.4. Binomial n 10, p 0.5, k 3: P(X = 3) = 0.1172, P(X ≤ 3) = 0.1719.
Three distributions,
exact arithmetic.
What each distribution models, how the probabilities are computed, and the discrete-vs-continuous trap.
Which one
Normal: a continuous measurement clustered around a mean with spread σ — heights, errors, averages of many things. Binomial: the number of successes in n independent yes/no trials with the same probability p. Poisson: the count of events in a fixed interval when they arrive at a steady rate λ. Mean and variance are μ and σ²; n·p and n·p·(1 − p); λ and λ.
Computing
The normal CDF is Φ(z) with z = (x − μ)/σ, via an erfc approximation accurate to about 10⁻⁷; the quantile inverts it with Acklam’s approximation plus one refinement step. Binomial and Poisson masses use the exact formulas through lnΓ so large n does not overflow; cumulative probabilities are summed term by term, not approximated.
Less-or-equal versus less-than
For a continuous variable P(X = x) is zero — the value shown is the density — and P(X ≥ x) equals P(X > x). For discrete counts they differ by the mass at k, so both are shown. When n·p and n·(1 − p) are both ≥ 10 the page notes that a normal approximation would be close, but still reports the exact sum. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- Normal via erfc (error < 1.2 × 10⁻⁷) and Acklam’s inverse with refinement; binomial C(n,k)p^k(1−p)^(n−k) and Poisson e^−λλ^k/k! computed through lnΓ; cumulative by exact summation
Last reviewed 21 September 2026. How results are checked: How we verify.