44. A random variable X has the Pareto distribution P( C, T) if its density is CT'/X" +I,O < T < X, 0 <

c. (i) Show that this defines a probability density. (ii) If X has distribution P( C, T), then Y = log X has exponential distribution Ea,

b) with = log T , b = l/c. (iii) If Xl' .. .' X" is a sample from P( C, T), use (ii) and Problem 3 to obtain UMP tests of

(a) H : T = TO against T"* TO when b is known ;

(b) H : C= co' T = TO against c> Co' T < TO '