31. 32. 33. 34. Let X1, X2 L, Xn be the values of a variable x. Define Xmax = max{X1, L, Xn}, n Xmin = min{X1, L, X}, R=x-x and s = (x-x)/n. max min R Show that 2n R 4 i=1 Let T be the minimum variance unbiased estimator (MVUE) of 0. Then prove that TK (Ka +ve integer) is the MVUE for E(T) provided E(T) < . Suppose (x1,y1).L.(xn, yn) represent a random sample from N2 (0,0,0,0,p). 102 Suppose p=Po (known), then find a confidence interval of /, with coefficient (1a) that incorporates the information that p=Po. Let X1, X2, L, X be i.i.d. with density confidence f(x, 0) == x>0, 0>0. 2 x" (a) (b) (c) Find MLE of 0 Derive the likelihood ratio test for Ho: 0-1 vs H : 01. If n=4 and the observations are X = 3.2, X2 = 4.0, X3 = 2.0, X4 = 5.6, find the P-value of the test derived in (b).