This post presents exercises on finding the probability distributions of order statistics to complement a discussion of the same topic.

Consider a random sample drawn from a continuous distribution with common distribution function . The order statistics are obtained by ranking the sample items in increasing order. In this post, we present some exercises to complement this previous post. The thought processes illustrated by these exercises will be helpful in non-parametric inference, specifically in the construction of confidence intervals for unknown population percentiles.

In the problems that follow, are the order statistics that arise from the random sample . There are two ways to work with the probability distribution of an order statistic . One is to find the distribution function . Once this is obtained, the density function is derived by taking derivative. Another way is to derive directly.

We assume that the random sample is drawn from a probability distribution with distribution function and with density function . To compute , note that for the event to occur, at least many sample items are less than . So the random drawing of each sample item is a Bernoulli trial with probability of success . Thus is the following probability computed from a binomial distribution.

Once the distribution function is found, the density function can be derived by taking derivative on . The density function can also be obtained directly by this thought process. Think of the density function as the probability that the th order statistic is right around . So there must be sample items less than and sample items above . One way this can happen is:

The first term is the probability that sample terms are less than . The second term is the probability that one sample item is right around . The third term is the probability that sample items are above . But this is only one way. To capture all possibilities, we multiply it by the multinomial coefficient.

_____________________________________________________________________________________

**Practice Problems**

*Practice Problems 1*

Draw a random sample of size 8 from the uniform distribution . Calculate the probability where is the third order statistic.

*Practice Problems 2*

Draw a random sample of size 5 from a continuous distribution with density function where . Find the probability that the sample median is less than 1.

*Practice Problems 3*

Draw a random sample of size 5 from the uniform distribution .

- Calculate the distribution function for the fourth order statistic. Then differentiate it to obtain the density function of .
- Use the thought process behind formula (2) above to directly write down the density function directly.

*Practice Problems 4*

Draw a random sample of size 9 from an exponential distribution with mean .

- Calculate the distribution function for the first order statistic (the minimum). Then differentiate it to obtain the density function of .
- Use the thought process behind formula (2) above to directly write down the density function directly.

*Practice Problems 5*

Draw a random sample of size 9 from an exponential distribution with mean .

- Calculate the distribution function for the second order statistic. Then differentiate it to obtain the density function of .
- Use the thought process behind formula (2) above to directly write down the density function directly.

*Practice Problems 6*

Draw a random sample of size 8 from the uniform distribution . Find , the expected value of the sixth order statistic.

*Practice Problems 7*

Draw a random sample of size 10 from the uniform distribution . Find , the variance of the ninth order statistic.

*Practice Problems 8*

Draw a random sample of size 6 from a population whose 25th percentile is 83. Find the probability the third order statistic is less than 83.

*Practice Problems 9*

Draw a random sample of size 6 from a population whose 75th percentile is 105. Find the probability the third order statistic is less than 105.

*Practice Problems 10*

Draw a random sample of size 4 from an exponential distribution with mean 10. Calculate .

*Practice Problems 11*

Draw a random sample of size 5 from a continuous distribution with density function where . Find , the expected value of the sample maximum.

*Practice Problems 12*

Draw a random sample of size 12 from an exponential distribution with mean 2. Calculate .

_____________________________________________________________________________________

_____________________________________________________________________________________

**Answers**

*Practice Problems 1*

*Practice Problems 2*

*Practice Problems 3*

*Practice Problems 4*

*Practice Problems 5*

*Practice Problems 6*

*Practice Problems 7*

*Practice Problems 8*

*Practice Problems 9*

*Practice Problems 10*

- = 0.756546693

*Practice Problems 11*

*Practice Problems 12*

_____________________________________________________________________________________