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Probability Distribution Calculator

Normal · Exponential · Uniform · Student's t · Chi-squared

Compute PDF, CDF, and quantiles for five continuous distributions. Interactive density curve with shaded probability region.

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CDF Value Table

CDF values F(x) = P(X ≤ x) for the currently selected distribution and parameters.

x	PDF f(x)	CDF F(x)	P(X > x)
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What Is a Probability Distribution?

A probability distribution is a mathematical description of the likelihood of each possible outcome of a random variable. When a random variable is continuous — capable of taking any value within a range — we describe it using a probability density function (PDF). Unlike discrete distributions (binomial, Poisson), which assign probabilities to individual outcomes, continuous distributions assign probabilities to intervals, calculated as the area under the PDF curve.

Understanding probability distributions is foundational in statistics, data science, machine learning, engineering, finance, and natural sciences. Every time you see a confidence interval, a p-value, or a hypothesis test, there is a distribution working behind the scenes.

Discrete vs. Continuous Distributions

A discrete distribution (binomial, Poisson, geometric) has a probability mass function (PMF) that assigns non-zero probability to specific integer values. For example, the number of heads in 10 coin flips can only be 0, 1, 2, ..., 10.

A continuous distribution can take any value in an interval or on the entire real line. The probability that a continuous variable equals any single exact value is zero — what matters is the probability over a range. This calculator focuses on five widely used continuous distributions.

PDF vs. CDF vs. Quantile Function

Concept	Notation	Definition
PDF	f(x)	Density at x; area under curve = 1
CDF	F(x) = P(X ≤ x)	Cumulative probability up to x
Survival	S(x) = 1 − F(x)	Probability of exceeding x
Quantile	Q(p) = F−¹(p)	x such that P(X ≤ x) = p

The Five Continuous Distributions Covered

Normal Distribution N(μ, σ²)

The normal (Gaussian) distribution is the most important distribution in statistics. Its symmetric bell-shaped curve is characterized by mean μ and standard deviation σ. The Central Limit Theorem guarantees that sums of many independent random variables converge to normality, making it applicable to measurement errors, heights, weights, test scores, and financial returns.

Exponential Distribution Exp(λ)

The exponential distribution models the waiting time between events in a Poisson process. With rate parameter λ, the mean waiting time is 1/λ. Its defining property is memorylessness: given that you have already waited t units, the distribution of additional waiting time is the same as the original. Used extensively in reliability engineering, queuing theory, and survival analysis.

Uniform Distribution U(a, b)

The uniform distribution assigns equal probability density to all values between a and b. It is the simplest continuous distribution — flat PDF, linear CDF. Used for random number generation, modeling outcomes where all values are equally likely (e.g., random point on a line segment), and as a prior in Bayesian statistics.

Student's t-Distribution t(ν)

The Student's t-distribution, introduced by William Gosset (pen name "Student") in 1908, is bell-shaped like the normal but with heavier tails controlled by degrees of freedom ν. It is used when estimating population means with small samples. As ν → ∞, t(ν) converges to N(0,1). Critical for t-tests, confidence intervals, and regression inference.

Chi-squared Distribution χ²(ν)

The chi-squared distribution with ν degrees of freedom is the sum of squares of ν independent standard normal variables. It is right-skewed and non-negative. Key applications include goodness-of-fit tests, tests of independence in contingency tables, and confidence intervals for population variance. As ν increases, it approaches a normal distribution by the CLT.

Connection to p-Values and Hypothesis Testing

A p-value is the tail probability from a test statistic under the null hypothesis. For a Z-test, the p-value is P(|Z| ≥ z_obs) using the normal distribution. For a t-test with ν degrees of freedom, it is P(|t| ≥ t_obs) using the t-distribution. For a chi-squared test with ν degrees of freedom, it is P(χ² ≥ χ²_obs) using the chi-squared distribution. This calculator computes all of these directly.

Distribution Summary Table

Distribution	Parameters	Mean	Variance	Use Case
Normal	μ, σ	μ	σ²	Natural measurements, CLT
Exponential	λ	1/λ	1/λ²	Waiting times, reliability
Uniform	a, b	(a+b)/2	(b−a)²/12	Equal-likelihood outcomes
Student's t	ν	0 (ν>1)	ν/(ν−2) (ν>2)	Small-sample inference
Chi-squared	ν	ν	2ν	Hypothesis tests, variance

Frequently Asked Questions

What is a probability distribution?

A probability distribution describes the likelihood of each possible outcome of a random variable. For continuous distributions, we compute probabilities over intervals using the area under a probability density function (PDF) curve. The total area under a PDF always equals 1, representing certainty that some outcome will occur.

What is the difference between PDF and CDF?

The PDF (Probability Density Function) f(x) gives the height of the distribution curve at each point — it represents relative likelihood, not a direct probability. The CDF (Cumulative Distribution Function) F(x) = P(X ≤ x) gives the probability that the random variable is at most x, computed as the integral of the PDF from −∞ to x. The CDF is always between 0 and 1 and is non-decreasing.

When should I use the normal distribution?

Use the normal distribution when your data is symmetric and bell-shaped around a central value. It's appropriate for measurement errors, heights and weights in large populations, standardized test scores, and many natural phenomena. More broadly, the Central Limit Theorem ensures that sample means follow approximately normal distributions for large samples, regardless of the underlying distribution.

What is the exponential distribution used for?

The exponential distribution models the time between events in a Poisson process — events that occur at a constant average rate λ. Common uses: time between customer arrivals at a queue, time until a light bulb fails, time between radioactive decay events, and inter-arrival times in network traffic. Its memoryless property means the remaining lifetime of a component is independent of how long it has already operated.

What is the Student's t-distribution?

The Student's t-distribution is a bell-shaped distribution with heavier tails than the normal, parameterized by degrees of freedom ν. It arises when estimating a population mean from a small sample with unknown variance: the test statistic (x̄ − μ) / (s/√n) follows a t-distribution with n−1 degrees of freedom. As ν → ∞, t(ν) converges to the standard normal N(0,1). At ν = 1, it becomes the Cauchy distribution.

What is the chi-squared distribution used for?

The chi-squared distribution χ²(ν) is the sum of squares of ν independent standard normal random variables. It is used in: (1) goodness-of-fit tests (does observed data fit an expected distribution?), (2) tests of independence in contingency tables, (3) confidence intervals for population variance, and (4) likelihood ratio tests. The test statistic is always non-negative and the distribution is right-skewed.

How do I find a p-value?

A p-value is the probability of observing a test statistic at least as extreme as yours, given that the null hypothesis is true. Steps: (1) compute your test statistic (Z, t, χ², etc.), (2) choose your test direction (left-tailed, right-tailed, or two-tailed), (3) use the CDF of the appropriate distribution. For a right-tailed test, p-value = P(X ≥ statistic) = 1 − F(statistic). Use the P(X > x) mode in this calculator to compute it directly.