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In sampling statistics (see Sec 18.7), we deal with an iid class of random variables, where n is a prescribed positive integer known as the sample size. An observation of this classgives an n -tuple of numbers . As an extension of the extreme values in the case of two variables (Ex 2.11),it is often useful to define a random variable Y 1 whose value for any ω is the smallest of the ; a second random variable Y 2 whose value at ω is the next smallest of the , and so on through Y n whose value at ω is the largest of the . We would like to be able to obtain the distributions for these new random variables in terms of the common distribution for the X i . We formulate the problem as follows.
Suppose is iid, with common distribution function F . Let
Then Y k is called the k th order statistic for the class . We wish to determine the distribution functions . Now, iff k or more of the X i have values no greater than t . We may view the process as a Bernoulli sequence of n trials. There is a success on the i th trial iff . The probability p of a success is . Hence
Remark . Once the common distribution function F for the X i is known, then the F k are calculated in a straightforward manner. For that purpose we may use the MATLAB function cbinom.
Suppose the X i are exponential (2). Then for positive t . Suppose . We calculate for .
n = 5;
t = 0.1:0.2:0.9;m = length(t);
F = 1 - exp(-2*t);for i = 1:m
FK(i,:) = cbinom(n,F(i),1:n);end
disp([t' F' FK]) % k = 1 k = 2 k = 3 k = 4 k = 5
0.1000 0.1813 0.6321 0.2249 0.0445 0.0046 0.00020.3000 0.4512 0.9502 0.7456 0.4091 0.1324 0.0187
0.5000 0.6321 0.9933 0.9354 0.7364 0.3946 0.10090.7000 0.7534 0.9991 0.9852 0.9000 0.6400 0.2427
0.9000 0.8347 0.9999 0.9968 0.9653 0.8064 0.4052 The following special case is important in characterizing the Poisson process (see Sec 21.1).
Order statistics for uniformly distributed random variables
Suppose is iid, uniform on . Determine the distribution functions for the order statistics.
SOLUTION
The common distribution function for the U i is given by . According to the result in Ex 2.16, the k th order statistic Y k has the distribution function
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