Section A
Q1.
(a) Consider a Weibull distribution with scale parameter α, location parameter μ = 0, and shape parameter γ. Show that its hazard rate is constant when γ = 1 and increases with time when γ = 2. What happens when γ > 2?
(b) The following are the number of defectives found on items produced in 15 days: 2, 3, 1, 2, 2, 1, 3, 2, 2, 1, 2, 2, 1, 0 and 0. Construct a control chart for this process and comment on whether the process is in control. (Use graph sheet provided)
(c) A forest department conducted a two-year testing of a new brand of an insecticide on ten plants. The survival times in months are recorded and are given below. The + symbol next to an observation signifies that the observation is censored (either removed or dropped from study). 24+, 16+, 8, 19, 10, 8+, 5, 17, 20, 10. Obtain an estimate of S(t) by Kaplan-Meier method and plot it against t. What is the estimated probability of 15 month survival? (Use graph sheet provided).
(d) Solve the two-person zero-sum game, whose pay-off matrix is given below, by graphical method. \beginbmatrix 3 & 5 \ -1 & 6 \ 4 & 1 \ 2 & 2 \ 1 & -3 \endbmatrix (Use graph sheet provided)
(e) Let 0.35, 0.69, 0.05, 0.87, 0.43 be five simulated values of uniform (0, 1) random variable. Using these, simulate values of a random variable X whose probability mass function is \beginarrayc|cccc x & 1 & 2 & 3 & 4 \ p(x) & 0.25 & 0.35 & 0.30 & 0.10 \endarray
