Discrete distributions formulas

Applies when each of the n consecutive integers is equally likely; zero elsewhere.

Here n = b - a + 1 and the values are consecutive integers. The variance depends only on n, not on where the block starts.

Special case a = 1, b = n. For a fair die (n = 6): mean 3.5, variance 35/12.

One trial, success probability p, 0 <= p <= 1.

Because X^2 = X for a 0/1 variable, E[X^2] = p, so Var(X) = p - p^2. The variance is largest at p = 1/2, where it equals 1/4.

Requires fixed n, independent trials and the same p on every trial. The probabilities sum to 1 by the binomial theorem.

Follows by adding n independent Bernoulli(p) variables: means add, and variances add because of independence. When 0 < p < 1 the variance is always less than the mean.

E[X^2] = Var(X) + (E[X])^2. Useful when a question asks for E[X^2] or the expectation of a quadratic in X.

Use it to compute successive probabilities quickly and to locate the mode.

Valid for 0 < p < 1. At the end points the distribution is degenerate: the mode is 0 when p = 0 and n when p = 1.

Both independence and the common p are essential. With different success probabilities the sum is in general not binomial, although means still add, and variances still add because of independence.

Independent trials, constant p with 0 < p <= 1. X counts the trial on which the first success occurs, so X >= 1. Mean 1/p and variance (1-p)/p^2. If instead the count is the number of failures before the first success, the PMF is (1-p)^k p for k = 0, 1, 2, ..., the mean is (1-p)/p and the variance is still (1-p)/p^2.