If Z1 and Z2 be two independent standard normal variables, show that the distribution of the ratio Z1 / |Z2| is Cauchy with probability density function f(x) = 1 / (pi (1 + x^2)), -inf < x < inf

2Medium10 marks

If X1 = Y1 + Y2, X2 = Y2 + Y3, X3 = Y3 + Y1, where Y1, Y2 and Y3 are uncorrelated random variables and each of which has zero mean and unit standard deviation, find the multiple correlation coefficient between X1 and X2, X3.

3Medium10 marks

Let F be a probability distribution function of a discrete random variable. Prove that the functions F^n and 1 - (1 - F)^n are also probability distribution functions.

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