Let X1, X2, X3,. . . , Xn are independent random variables distributed uniformly over the interval (0, 1). n is any natural number, fixed.

(a) We define Yi = a + (b - a) Xi, for any real numbers a and b with b> a. The random variables Yi are also independent and identically distributed. Find

FY (t) = P {Y1

the distribution function of Yi. What is the probability distribution of Yi (give his name and its parameters)?

(b) We define Z = max {Yi : 1?i?n} and its distribution function FZ (t) = P {Z?t}. Explain why

1. 

n fz(t) = (b - a )n (t - a ) n-1 ( a, b) (t ).\f