Section A
Q1.
(a)
For n events A1, A2, . . , An, show that
Pleft[igcap_{i=1}^{n} Ai
ight] geq sum_{i=1}^{n} P(Ai) - (n-1).
Let {An} be an increasing sequence of sets (events), then show that
lim_{n o infty} P(An) = Pleft(lim_{n o infty} An
ight) = Pleft(igcup_{n=1}^{infty} An
ight). .
(b)
Let Omega = {1, 2, 3, 4} and pi = P(i), i = 1, 2, 3, 4.
Assume that
p1 = rac{sqrt{2}}{2} - rac{1}{4},
p2 = rac{1}{4},
p3 = rac{3}{4} - rac{sqrt{2}}{2}, and
p4 = rac{1}{4}.
Define the events
E1 = {1, 3}, E2 = {2, 3} and E3 = {3, 4}.
Check whether E1, E2 and E3 are mutually independent.
(c) Suppose that X1, . . , Xn form a random sample from a Uniform distribution on the interval [ heta1, heta2] where both heta1 and heta2 are unknown (-infty < heta1 < heta2 < infty). Find the maximum likelihood estimators of heta1 and heta2.
(d) Suppose that X1, . . , Xn form a random sample from a Gamma distribution for which the value of parameter lpha is unknown (lpha > 0) and value of parameter eta is known. Show that the joint probability density function of X1, . , Xn has a Monotone Likelihood Ratio (MLR).
