Interactive Tools
Probability distributions — interactive.
Click any distribution, slide its parameters, and watch the PMF/PDF and CDF respond live. Built for JEE, NEET, B.Sc/M.Sc statistics, and data-science learners.
Filter
Use case
CLT simulation and hypothesis test visualization.
Notation
PMF — Probability Mass Function. For discrete distributions: gives P(X = k), the probability that the random variable X takes the exact value k.
PDF — Probability Density Function. For continuous distributions: f(x) is the density, not a probability. Probabilities are obtained by integrating: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.
CDF — Cumulative Distribution Function. F(x) = P(X ≤ x). Defined the same way for both discrete and continuous distributions; rises from 0 to 1.
PDF — Probability Density Function. For continuous distributions: f(x) is the density, not a probability. Probabilities are obtained by integrating: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.
CDF — Cumulative Distribution Function. F(x) = P(X ≤ x). Defined the same way for both discrete and continuous distributions; rises from 0 to 1.
Symbols and Greek letters
μ (mu) — population mean
σ (sigma) — population standard deviation
σ² — population variance
λ (lambda) — rate parameter (Poisson, Exponential)
ν (nu) — degrees of freedom (Student's t)
k — degrees of freedom (Chi-squared); also count in PMF
d₁, d₂ — numerator and denominator df (F)
n, p — number of trials, success probability (Binomial)
π (pi) ≈ 3.14159
e ≈ 2.71828 — base of natural log
Γ(·) — gamma function: Γ(n) = (n−1)! for integer n
B(α,β) — beta function = Γ(α)Γ(β)/Γ(α+β)
ⁿCₖ — binomial coefficient "n choose k" = n!/(k!(n−k)!)
χ² — chi-squared (Greek "chi" squared)
α, β — shape parameters (also significance level / power complement)
Z — standardized variable: Z = (X − μ)/σ ~ N(0, 1)
How to use it
Pick, slide, understand.
1
Pick a distribution
Filter by family or use case (discrete/continuous, count, time, ratio, …) and click any card to pin it.
2
Slide the parameters
Move the sliders for n, p, λ, μ, σ — the PMF/PDF, CDF, and live statistics update instantly.
3
Read the formulas
Probability, mean, variance, skewness, kurtosis — exact closed-form expressions visible alongside the plots.
4
Run a probability
Use the inline calculator: P(X ≤ x), P(a ≤ X ≤ b), P(X > x) — checked instantly against the formulas.
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