![]() In descriptive statistics, we define the sample percentiles using the order statistics (even though the term order statistics may not be used in a non-calculus based introductory statistics course). Then, the following shows the pdf of the order statistic of the uniform distribution on the unit interval and its mean and variance: ![]() In general, the pdf of a beta distribution and its mean and variance are: The above density function is from the family of beta distributions. Since the distribution function of is where, the probability density function of the order statistic is: Suppose that the random sample are drawn from. The Order Statistics of the Uniform Distribution ![]() In the previous post The distributions of the order statistics, we derive the probability density function of the order statistic: Since this is random sampling from a continuous distribution, we assume that the probability of a tie between two order statistics is zero. In other words, is the smallest item in the sample and is the second smallest item in the sample and so on. The order statistics are obtained by ordering the sample in ascending order. Suppose that we have a random sample of size from a continuous distribution with common distribution function and common density function. For a discussion on the distributions of order statistics of random samples drawn from a continuous distribution, see the previous post The distributions of the order statistics. We also present an example of using order statistics to construct confidence intervals of population percentiles. This leads to a discussion on estimation of percentiles using order statistics. In this post, we show that the order statistics of the uniform distribution on the unit interval are distributed according to the beta distributions.
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