What is uniform distribution?

Uniform distribution is a probability distribution where all outcomes have equal likelihood of occurring.

In probability theory and statistics, a uniform distribution is a probability distribution where all outcomes have equal likelihood of occurring. This means that the probability of any given outcome is the same as the probability of any other outcome. The uniform distribution is often used to model situations where all outcomes are equally likely, such as rolling a fair die or selecting a card from a well-shuffled deck.

The probability density function (PDF) of a continuous uniform distribution is given by:

f(x) = 1/(b-a) for a ≤ x ≤ b

where a and b are the lower and upper bounds of the distribution, respectively. The cumulative distribution function (CDF) of a continuous uniform distribution is given by:

F(x) = (x-a)/(b-a) for a ≤ x ≤ b

The mean and variance of a continuous uniform distribution are given by:

μ = (a+b)/2

σ^2 = (b-a)^2/12

In a discrete uniform distribution, the probability of each outcome is also equal. For example, if we roll a fair six-sided die, the probability of rolling any given number is 1/6. The probability mass function (PMF) of a discrete uniform distribution is given by:

P(X=x) = 1/n for x = 1, 2, ..., n

where n is the number of possible outcomes. The mean and variance of a discrete uniform distribution are given by:

μ = (n+1)/2

σ^2 = (n^2-1)/12

Overall, the uniform distribution is a simple and useful probability distribution that is often used in modelling situations where all outcomes are equally likely.