Uniform distribution

Definition

A uniform distribution is a type of probability distribution in which all outcomes are equally likely. In a continuous uniform distribution, every interval of the same length within the distribution's range has an equal probability of occurring.

5 Must Know Facts For Your Next Test

1. The probability density function (PDF) for a continuous uniform distribution is $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$.
2. The mean (expected value) of a continuous uniform distribution is $(a + b)/2$.
3. The variance of a continuous uniform distribution is $(b - a)^2 / 12$.
4. Uniform distributions are used to model scenarios where each outcome in an interval is equally likely.
5. Using the Central Limit Theorem, the sampling distribution of the sample mean from a large sample size drawn from any uniformly distributed population will approximate a normal distribution.

Review Questions

• What is the formula for the probability density function (PDF) of a continuous uniform distribution?
• How do you calculate the mean and variance of a continuous uniform distribution?
• Explain how the Central Limit Theorem applies to samples drawn from a uniformly distributed population.

Related terms

Central Limit Theorem:

A statistical theory stating that, given a sufficiently large sample size from any population with finite variance, the sampling distribution of the sample mean will tend to be approximately normally distributed.

Probability Density Function:

A function that describes the likelihood of a random variable taking on different values; for continuous variables, it represents probabilities over intervals.

Variance:

A measure of how much values in a data set differ from the mean; calculated as the average squared deviation from the mean.