Answer Happy

b8 Consider a parallel system of two independent unreliable machines. What is the probability that machine 1 fails before machine 2 if the life time X of machine i is exponential with parameter 5. i = 1.2? Also draw the integral area. Explain your steps. (30 pts) b9 To test a proportion of defective items one draws a sample X = (X1,..., X.) from a Bernoulli population with an unknown 0 € (0,1). (6) Give the likelihood function of the sample. (ii) Find the m.l.e. and MLE of 0. Explain your steps. (30 pts) RU b10 Find the mean residual time (t) = E(X – {|X > 1) of a serial system of two independent and identically distributed lifetimes of its components, each having the common exponential distribution with parameter 1. Hint. Use formula (*) r(t) = to SR(u)du. (30 pts) bu Let f(x,y) = { 12., 0<r<y<1 0, elsewhere be the joint pdf of r.v.'s (X,Y). Find the conditional pdf f(x|y). Explain your steps. (25 pts)