Question

1. [30pt] Suppose x is uniformly distributed over (-2, 1] and yx 1. Find the pdf , 2. [30pt] Let xi be 1.1 d 1 rend.

Answer 1

"We know from second shifting property of Laplace transform is ""[t g(t)] = G(s) then L[u(t - a) g(t -a)] = e^- as G(s)"" Here we have to fixed inverse Laplace transform of F(s) = e^-3s(1 - 5s)/s^2 + 16 According to above def G(s) = 1 - 5s/s^2 + 16 and g(t) is the inverse Laplace of G(s) So L^- 1 (1 - 5s/s^2 + 16) = L^- 1(1/s^2 + 16 - 5s/s^2 + 16) = L^- 1(1/s^2 + 16) - L^- 1(5s/s^2 + 16) = 1/4 sin(4t) - 5 cos(4t) and inverse Laplace of F(s) = 4(t - 3)(1/4 sin (4(t - 3)) - 5 cos (4 (t - 3)) Therefore L^- 1(e^- 3s (1 - 5s)/s^2 + 16) = 1/4{(4(t - 3))sin (4(t - 3) - 20 cos (4(t - 3))}"