uniform distribution

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.

• 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.

• Quantile-Quantile (q-q) Plots

  • If the spinner is fair, then these numbers should follow a uniform distribution.
  • Based upon our description of the spinner, we expect a uniform distribution to model these data.
  • Just as in the case of the uniform distribution, we have 5 intervals.
  • A physical device that gives samples from a uniform distribution.
  • The empirical and theoretical cumulative distribution functions of a sample of 100 uniform points

• 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.