Section A
Q1.
(a)
Let P(An) = 0 for each n geq 1. With proper justification show that
Pleft(igcup_{n=1}^{infty} An
ight) = 0.
(a)
Let P(An) = 1 for each n geq 1. With proper justification show that
Pleft(igcap_{n=1}^{infty} An
ight) = 1.
(b)
Do the following functions define cumulative distribution functions ?
(i)
F(x) = egincases 0 & extif x < 0, \ x & extif 0 leq x < rac12, \ 1 & extif x geq rac12. endcases
(b)
(ii)
F(x) = egincases 0 & extif x leq 1, \ 1 - rac1x & extif x > 1. endcases
(c)
Construct sequential probability ratio test for testing H0 : p = p0 against H1: p = p1, when X is a Bernoulli random variable defined with probability function given by
f(x; p) = p^{x} (1-p)^{1-x}, x: 0, 1; 0 < p < 1.
(d)
A company wishes to purchase one of five different machines : A, B, C, D or E. In an experiment designed to determine whether there is a performance difference among the machines, five experienced operators were assigned to work on the machines for equal times. Data given below represents the number of units produced by each machine : A 68 72 77 42 53
B 72 53 63 53 48
C 60 82 64 75 72
D 48 61 57 64 50
E 64 65 70 68 53
Test the hypothesis that there is no difference between the machines at 0.05 significance level.
[Note : Refer the chi-square table for the theoretical value of \chi2].
Chi-square distribution table is provided at the end of the booklet.
