37. Consider a linear model Y = X0+ where Y is a 41 vector of observations, =(0,0,0) is a vector of unknown parameters, 1 -1 0 100 X4x3= 1 0 1 1 00 and & is a 41 vector of uncorrelated random errors with mean 0 and variance o. (a) (b) Verify whether the following parametric functions are estimable (i) 01+ 02, (ii) 01+ 02 +03 Find the best linear unbiased estimator(s) of the estimable parametric function(s) in (a) above and obtain the variance of the estimator(s). 38. suppose tips - thon:" )-w,[[4), - %. with 11 > 0. 21 22 39. 40. (P21) % Prove that the necessary and sufficient condition for x and x2 to be independent is 12=0. You may assume x ~ N (), i=1,2. Pi 0/ Suppose the problem is to classify an observation x into one of the populations P, i=1,2. Suppose f(x) denotes the density of x corresponding to population Pi. Also we attach the prior probability pi (i = 1, 2) for an observation x to belong to population Pi. Find the total probability of misclassification (TPM) and prove that the classification rule minimizing TPM is given by: for an x, if f(x) P2 classify it as an observation belonging to population P, and f(x) P otherwise belonging to population P2. An unknown number N of taxis plying in a town are supposed to be serially numbered from 1 to N. If the n different taxis you have come across in the town can be assumed to form a simple random sample with replacement, find an unbiased estimator of the total number of taxis in the town. Also find the variance of your estimator.