Iff (t) is the unconditional density of the time of failure of a system and h(t) is the hazard rate (or conditional failure rate or conditional density of T, given T > t) find f (1) in terms of h(t). Deduce that T follows a Rayléigh distribution, when h(t) = t.

Let X1, . . . , Xn ∼ iid Unif(0, θ). (a) Derive an asymptotic distribution for the MOM estimator ˜θ = 2X¯ of the form.(b) From this choose an approximate pivot and interval to get an interval that asymptotically has100(1 − α)% coverage.

C, D, and E

2. A particle of mass, m, has the wavefunction given by: (x, t) = Ce-a[(mx²/h) + it] . (a) Sketch Re [(x,0)] as a function of x. (b) Sketch Re [p(0, t)] as a function of t. (c) Sketch the probability density for finding the particle at x at t = 0. (d) Explain the time dependence in the probability density for locating the particle in space. (e) Find C (f) Find (x) (g) Find (x2) (h) Find ox (i) Find (p) (j) Find (p?) (k) Find op (1) Is the product, O,0, consistent with the Heisenberg uncertainty principle? (m) Find T