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(1) Let Y1, Y2 ...Yn be independent uniformly distributed random variables of the interval [0, θ ]. Find the (a) Cumulative Distribution Function (CDF ) of Y(n)= max (Y1,Y2, ,...,Yn ). (b) Probability density function of ( pdf ) Y(n). (2) Let Y1, Y2, ,..., Y15 be independent random variables having exponential distribution with mean 10. Give the density function for Y(8), the sample median of these data. (3) Ten numbers are generated at random between 0 and 1. Write the joint density of the 5th and 6th order statistics. (1) Suppose that X1,X2 X n and Y1,Y2 Yn are independent random samples means from populations with 14 and 12 and variances of and 02, respectively. Show that the random variable ( X - Y)-(44-12) U n 1 n n Satisfies the conditions of Theorem 7.4 and thus that the distribution function of n converges to a standard normal distribution function as n 8. [Hint: Consider W, = Xi -Yi, for (2) The result in exercise (1) holds even if the sample sizes differ. That is, if X 1, X 2 , Xn, and Y1YY Yn, constitute independent random samples from populations with means M and 112 and variances of and 52, respectively, then X-Y - will be approximately normally distributed, of 2 2 O2 for large n and n2, with mean M1-M2 - and variance + n n2 The flow of water through soil depends on, among other things, the porosity (volume proportion of voids) of the soil. To compare two types of sandy soil, n = 50 measurements are to be taken on the porosity of soil A and n2 = 100 measurements are to be taken on soil B. Assume that of = 0.01 and o2 = 0.02. Find the probability that the difference between the sample means will be within 0.05 unit of the difference between the population means - (3) Refer to exercise (2). Suppose that n = n2 = n, , find the value of n that allows the difference between the sample means to be within 0.04 unit of M - 12 with probability 0.90.

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