Probability Distribution Calculator

A probability distribution assigns probability to every value a random variable can take. The form depends on whether the variable is discrete or continuous.

Discrete — probability mass function (PMF) $p(x) = P(X = x)$. It is valid if and only if both conditions hold:

$p(x) \ge 0 \text{ for all } x \qquad \text{and} \qquad \sum_{\text{all } x} p(x) = 1$

Continuous — probability density function (PDF) $f(x)$, valid when

$f(x) \ge 0 \qquad \text{and} \qquad \int_{-\infty}^{\infty} f(x)\,dx = 1$

For a PDF, $f(x)$ is not a probability — it is a density, and it may exceed $1$. Probability is area: $P(a \le X \le b) = \int_a^b f(x)\,dx$, which also means $P(X = a) = 0$ for any single point.

The cumulative distribution function (CDF) is $F(x) = P(X \le x)$ in both cases: a running sum for a PMF, an integral for a PDF.

Discrete case

$\mu = E[X] = \sum x \, p(x) \qquad \sigma^2 = \sum (x - \mu)^2 p(x) = E[X^2] - \mu^2$

The right-hand form $E[X^2] - \mu^2$ is the computational shortcut: build one column of $x\,p(x)$ and one of $x^2 p(x)$, total both, then subtract $\mu^2$. The standard deviation is $\sigma = \sqrt{\sigma^2}$.

Continuous case

$\mu = \int x f(x)\,dx \qquad \sigma^2 = \int x^2 f(x)\,dx - \mu^2$

Building a distribution from scratch

1. List every distinct value the variable can take — the values must be exhaustive and mutually exclusive.
2. Assign each one its probability, usually by counting outcomes.
3. Verify the total is exactly $1$. If a probability is unknown, that equation solves for it.
4. Only then compute $\mu$, $\sigma^2$, or any $P(X \ge k)$.

A valid distribution requires no assumption about normality or sample size — those conditions belong to inference, not to the distribution itself.

• Forgetting to check the sum. A table that totals $0.95$ or $1.05$ is not a distribution, and every mean or variance computed from it is meaningless.
• Allowing a negative probability. A formula such as $p(x) = (x-3)/6$ may sum to $1$ yet still produce a negative value; both conditions must hold, not just one.
• Treating a density as a probability. For a continuous variable $f(2) = 1.5$ is perfectly legal. Only areas are probabilities, so always integrate.
• Using $\sum (x - \mu)^2 / n$. That is the formula for raw data. For a distribution you weight by $p(x)$, not by $1/n$.
• Subtracting $\mu$ before squaring in the shortcut. The identity is $E[X^2] - \mu^2$, never $E[X^2 - \mu^2]$ computed halfway.
• Mixing up $P(X \ge 2)$ and $P(X > 2)$. For a discrete variable these differ by the whole of $p(2)$; for a continuous one they are equal.