uniform distribution

(noun)

a family of symmetric probability distributions such that, for each member of the family, all intervals of the same length on the distribution's support are equally probable

Related Terms

Examples of uniform distribution in the following topics:

  • The Uniform Distribution

    • The continuous uniform distribution is a family of symmetric probability distributions in which all intervals of the same length are equally probable.
    • Many programming languages have the ability to generate pseudo-random numbers which are effectively distributed according to the uniform distribution.
    • If $u$ is a value sampled from the standard uniform distribution, then the value $a+(b-a)u$ follows the uniform distribution parametrized by $a$ and $b$.
    • The uniform distribution is useful for sampling from arbitrary distributions.
    • Contrast sampling from a uniform distribution and from an arbitrary distribution

  • Practice 1: Uniform Distribution

    • The age of cars in the staff parking lot of a suburban college is uniformly distributed from six months (0.5 years) to 9.5 years.

  • The Uniform Distribution

    • The previous problem is an example of the uniform probability distribution.
    • Illustrate the uniform distribution.
    • We will assume that the smiling times, in seconds, follow a uniform distribution between 0 and 23 seconds,inclusive.
    • The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution.
    • The notation for the uniform distribution is X∼U(a,b) where a = the lowest value of x and b = the highest value of x.

  • Species Distribution

    • Individuals of a population can be distributed in one of three basic patterns: they can be more or less equally spaced apart (uniform dispersion), dispersed randomly with no predictable pattern (random dispersion), or clustered in groups (clumped dispersion) .
    • Uniform dispersion is observed in plant species that inhibit the growth of nearby individuals.
    • Animals that maintain defined territories, such as nesting penguins, also exhibit uniform dispersion.
    • A population may have a uniform, random, or clumped distribution.
    • Territorial birds, such as penguins, tend to have uniform distribution.

  • Lab 1: Chi-Square Goodness-of-Fit

    • The student will evaluate data collected to determine if they fit either the uniform or exponential distributions.
    • Test to see if grocery receipts follow the exponential distribution with decay parameter 1/x .
    • Did your data fit either distribution?
    • In general, do you think it's likely that data could fit more than one distribution?

  • Properties of Continuous Probability Distributions

    • The graph of a continuous probability distribution is a curve.
    • There are many continuous probability distributions.
    • In this chapter and the next chapter, we will study the uniform distribution, the exponential distribution, and the normal distribution.
    • The following graphs illustrate these distributions.
    • The graph shows a Uniform Distribution with the area between x=3 and x=6 shaded to represent the probability that the value of the random variable X is in the interval between 3 and 6.

  • Continuous Probability Distributions

    • A continuous probability distribution is a probability distribution that has a probability density function.
    • There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.
    • This requirement is stronger than simple continuity of the cumulative distribution function, and there is a special class of distributions—singular distributions, which are neither continuous nor discrete nor a mixture of those.
    • An example is given by the Cantor distribution.
    • For example, the uniform distribution on the interval $\left[0, \frac{1}{2}\right]$ has probability density $f(x) = 2$ for $0 \leq x \leq \frac{1}{2}$ and $f(x) = 0$ elsewhere.

  • Review

    • The distribution for X is Uniform.
    • The distribution for $\bar{X}$ is still Uniform with the same mean and standard dev. as the distribution for X.
    • The distribution for X is uniform.
    • The distribution for ∑ Xis still uniform with the same mean and standard deviation as the distribution for X.
    • The distribution for ∑ X is Normal with the same mean but a larger standard deviation as the distribution for X.

  • Sampling Distribution of the Mean

    • The sampling distribution of the mean was defined in the section introducing sampling distributions.
    • The parent population was a uniform distribution.
    • You can see that the distribution for N = 2 is far from a normal distribution.
    • For N = 10 the distribution is quite close to a normal distribution.
    • The parent population is uniform.

  • Sampling Distributions and the Central Limit Theorem

    • The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution.
    • For large enough $n$, the distribution of $S_n$ is close to the normal distribution with mean $\mu$ and variance
    • The upshot is that the sampling distribution of the mean approaches a normal distribution as $n$, the sample size, increases.
    • The usefulness of the theorem is that the sampling distribution approaches normality regardless of the shape of the population distribution.
    • The sample means are generated using a random number generator, which draws numbers between 1 and 100 from a uniform probability distribution.