Q1.
(a) 1. The lifetime of a mobile charger (in hours) has the normal distribution with mean (μ) = 100 and variance (σ^2) = 400.
(i) What is the probability that the mobile charger lasts at least 125 hours?
(ii) If the mobile charger has already lasted for 105 hours, what is the conditional probability that it will last another hours? (Normal Distribution Table is given in Page Nos. and 11)\n
(a)
1. The lifetime of a mobile charger (in hours) has the normal distribution with mean (μ) = 100 and variance (σ^2) = 400.
(i) What is the probability that the mobile charger lasts at least 125 hours?
(ii) If the mobile charger has already lasted for 105 hours, what is the conditional probability that it will last another hours? (Normal Distribution Table is given in Page Nos. and 11)\n
(b)
State Lindeberg condition for non-identically distributed independent variables to hold central limit theorem (CLT).
(ii) Examine whether CLT holds for the sequence Xn, where PXn = ± (1/2n) = 1/2\n
(c) observations from a location parameter family with cumulative distribution function F(x - θ), -∞ < θ < ∞.
(d)
Show that R = X_(n) - X_
(1) is ancillary statistic, where X_(n) = maxi Xi and X_
(1) = mini Xi.\n Suppose H0: θ = 1 versus H1: θ = 1/2, where θ is the mean of a Poisson random variable. Let X and Y be a random sample from Poisson(θ) distribution. Consider the following test procedure: Reject H0 if X = 1 or (Y = 1 and X + Y ≤ 2), otherwise accept H0.
(e) Determine the probability of type I and type II errors. In an ecological study of the feeding behaviour of birds, the number of hops between flights is counted for several birds: No. of hops Observed frequency 1 48 2 31 3 20 4 9 5 6 6 5 7 2 8 9 1 10 1 11 0 12 0 Total 130. Assuming that the data are generated by a geometric (p) model and take a uniform prior for p, what is the posterior distribution of parameter p? What are the mean and the standard deviation of the posterior distribution?
