iid 1. Let Y1,..., Yn * Bernoulli(p). It can be shown using the properties of the MLE that when n is sufficiently large: yn(Ỹ – p) _d, N (0, 1) VÝ(1 – Ý) (a) Use this result to show that: - p) S Za/2 VÝ(1 P 21– a Y) (b) Use your answer in the previous part to derive an approximate 1 – a confidence interval for p.

2.1 Determine y(t) by deriving the Inverse Laplace transform: y(t)=L 98² (s+1)(s²+2s+10) (8) 2.2 Determine L (In t). (7) 2.3 Illustrate the convolution of independent binomial distributions of X and Y to determine the distribution of Z = X+Y. X is distributed b(n, p) and Y is distributed, independently of X, b(m, p). (4) 2.4 Using convolutions determine the distribution of X + Y if X is distributed negative binomial (r,p) and Y is distributed negative binomial (s,p), 0

2.8 Consider the triangular random variable with pdf 0 f(x) = X X 2a (b − a)² (2b − x) (b − a)² if x 2b, if 2a < x

1 Find the domain of r(t) = Vt² – 9i – j+ In(1 – t)k. t + 4 %3D O (-0, -4) U (-4, -3) O [-3, 1) O [3, 0) O (-00, -4) U (-4, -3] O 0, the empty set. (No real numbers)