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 1-A |
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 1-B |
As in Problem 1-A, 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 1-C |
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 1-D |
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 1-E |
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 1-F |
Suppose that is a random sample drawn from a uniform distribution on the interval . Let be the sample range.
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Practice Problem 1-G |
As in Problem 1-F, 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 .
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Practice Problem 1-H |
Suppose that is a random sample drawn from a uniform distribution on the interval . Let , and be the resulting order statistics.
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Practice Problem 1-I |
Suppose that is a random sample drawn from an exponential distribution with mean . Let , and be the resulting order statistics.
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Practice Problem 1-J |
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.
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Practice Problem 1-K |
As in Problem 1-J, 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.
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Problem | ………..Answer |
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1-A |
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1-B |
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1-C |
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1-D |
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1-E |
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1-F |
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1-G |
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1-H |
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1-I |
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1-J |
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1-K |
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Tagged: Order statistics
[…] The multinomial approach we use here is discussed in this previous post. The only difference is that the random samples discussed here are drawn from the lognormal distribution. A practice problem set for the multinomial approach is found here. Another set of practice problems for order statistics is found here. […]