Question

) Y.Ya..., Y, form a random sample from a probability distribution with cumu- lative distribution function Fy (u) and probability density function fr(u). Let Write the cumulative distribution function for Ya) in terms of Fy(y) and hence show that the probability density function for Yy is fy(1)(y) = n(1-Fr (v))"-ify(y). [8 marks] (b) An engineering system consists of 5 components connected in series, so, if one components fails, the system fails. The lifetimes (measured in years) of the 5 components, Y, Ya, . . . ,%, are all independent and identically distributed. ppose the lifetimes follow the standard uniform distribution U(0,1). Find the probability density function for Ya), the time to failure for the system, and hence find the probability that the system functions for at least 6 [10 marks] mopths without failing. irf, instead, the lifetimes follow an exponential distribution with mean then Y(1) follows an exponential distribution with mean θ/5. Prove this result. Assuming that the only information available is a single observation on Ya) find the most powerful test of size 0.05 for H0 : θ 6, versus Hi : θ :-82, where 81 < 02. (Hint: the probability density function and cumulative distribution function for an exponential random variable with mean 8 are f(v) -1exp(-y/0), y > 0, and F(y) 1- exp(-y/0), y 0, respec- 12 marks tively.)

I can do the first part of the question 1a, could someone show me step by step how to do do 1b?

Answer 1