Probability Distributions Explorer

Select a distribution, adjust parameters, and visualize probability density or cumulative distribution functions. Drag the bound handles or type values to compute probabilities. All calculations run in your browser.

Distribution

Mean (μ)

Std Dev (σ)

Probability Density Function (PDF)

Drag the orange handles to adjust the probability region

Probability Calculation

z = -1.0000

≤ X ≤

z = 1.0000

) =0.6827

Inverse Lookup (Normal Only)

Find x where P(X ≤ x) =

Distribution Properties

Mean

0.0000

Variance

1.0000

Skewness

0.0000

Excess Kurtosis

0.0000

Reference Guide

PDF vs CDF

The probability density function (PDF) gives the relative likelihood of a continuous random variable taking a specific value. The area under the curve between two points gives the probability of falling in that range.

Continuous probability

P(a \le X \le b) = \int_a^b f(x)\,dx

The cumulative distribution function (CDF) gives the probability that a variable takes a value less than or equal to x. It is the running integral of the PDF.

CDF definition

F(x) = P(X \le x) = \int_{-\infty}^x f(t)\,dt

For discrete distributions, the probability mass function (PMF) gives $P(X = k)$ directly, and the CDF is the running sum $F(x) = \sum_{k \le x} P(X=k)$ .

The Normal Distribution

The bell-shaped curve parameterized by mean $\mu$ and standard deviation $\sigma$ . Many natural measurements follow this distribution.

PDF

f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

The 68-95-99.7 rule About 68% of values fall within

1\sigma

of the mean, 95% within

2\sigma

, and 99.7% within

3\sigma

Z-scores Standardize any normal variable via

z = (x - \mu) / \sigma

Discrete Distributions

Binomial Counts successes in

n

independent trials, each with success probability

p

P(X = k) = \binom{n}{k}\,p^k\,(1-p)^{n-k}

Poisson Counts events in a fixed interval when events occur independently at average rate

\lambda

P(X = k) = \frac{\lambda^k\,e^{-\lambda}}{k!}

For discrete distributions, probabilities are computed by summing the PMF over all integers in the range.

Expected Value and Variance

The mean (expected value) is the center of the distribution. Variance measures how spread out values are around the mean. Each distribution has its own formulas.

Normal

E[X]=\mu

\text{Var}(X)=\sigma^2

Binomial

E[X]=np

\text{Var}(X)=np(1-p)

Poisson

E[X]=\lambda

\text{Var}(X)=\lambda

Uniform

E[X]=\frac{a+b}{2}

\text{Var}(X)=\frac{(b-a)^2}{12}