Math calculators

Probability distributions

Pdf, cdf, interval probabilities and quantiles for the normal, Student t, lognormal, binomial, Poisson, exponential and uniform distributions.

Inputs

Results

Density

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Methodology

Conventions

  • For the binomial and Poisson the density is the probability of exactly x; interval probabilities include both ends.
  • The lognormal is parametrised by μ and σ, the mean and standard deviation of ln X.
  • The t, binomial and Poisson cdfs use the regularised incomplete beta and gamma functions; quantiles are found by inversion.

References

  • Press, Teukolsky, Vetterling and Flannery, Numerical Recipes, 3rd ed. (2007), ch. 6
  • NIST Digital Library of Mathematical Functions, dlmf.nist.gov, ch. 8

Formulas

Normal
f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x - \mu)^2 / 2\sigma^2}
Binomial
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
Poisson
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
Quantile
Q(p) = \inf\{x : F(x) \ge p\}