This post presents exercises on order statistics, reinforcing the concepts discussed in two blog posts in a companion blog on mathematical statistics.
The first blog post from the companion blog is an introduction to order statistics. That post presents the probability distributions of the order statistics, both individually and jointly. The second post presents basic examples illustrating how to calculate the order statistics.
Practice Problem 1A 
A random sample of size 4 is drawn from a population that has a uniform distribution on the interval . The resulting order statistics are , , and .
Determine the cumulative distribution function (CDF) of the 3rd order statistic . Evaluate the probability . 
Practice Problem 1B 
As in Problem 1A, a random sample of size 4 is drawn from a population that has a uniform distribution on the interval . The resulting order statistics are , , and . Evaluate the conditional probability . 
Practice Problem 1C 
The random sample of size 9 is drawn from a population that has a uniform distribution on the interval . Evaluate the mean and variance of the 7th order statistic . 
Practice Problem 1D 
Suppose that the random sample of size 2 is drawn from a population that has an exponential distribution with mean 10. Let be the sample minimum and let be the sample maximum. Evaluate the conditional probability . 
Practice Problem 1E 
Suppose that is a random sample drawn from an exponential distribution with mean 10. The sample median here is the 2nd order statistic . Evaluate the probability that the sample median is between 5 and 10. 
Practice Problem 1F 
Suppose that is a random sample drawn from a uniform distribution on the interval . Let be the sample range.

Practice Problem 1G 
As in Problem 1F, is a random sample drawn from a uniform distribution on the interval . Let be the sample range. The following relationship relates the variance of the sample range with the variances and covariance of and .

Practice Problem 1H 
Suppose that is a random sample drawn from a uniform distribution on the interval . Let , and be the resulting order statistics.

Practice Problem 1I 
Suppose that is a random sample drawn from an exponential distribution with mean . Let , and be the resulting order statistics.

Practice Problem 1J 
Suppose that is a random sample drawn from a continuous distribution with density function where . Let the resulting order statistics be , and where is the sample minimum, is the sample median and is the sample maximum.

Practice Problem 1K 
As in Problem 1J, suppose that is a random sample drawn from a continuous distribution with density function where . Let the resulting order statistics be , and where is the sample minimum, is the sample median and is the sample maximum.

Problem  ………..Answer 

1A 

1B 

1C 

1D 

1E 

1F 

1G 

1H 

1I 

1J 

1K 

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