Question

Section B (50 marks) Answer any two questions from this section. All questions carry equal marks. 1. (a) The density function of X is given by $(x) = { a + bx?, OSX31 0, otherwise If E[X] = , find i. the cumulative density function, F(X). [5 mark ii. P(x < [3 mark (b) Given the observations Ii, i = 1, 2 and their corresponding weight, W;, i = 1, show that the weighted average Xw = 221 + (1 - a)x2, where 0 <a<1. [4 mark i. Given that, wi = 3w2, find a. (4 mark ii. Hence or otherwise show that given equal weights, w, for all the observation Ti, i = 1, 2 the weighted mean, X, is equal to the arithmetic mean, X. [4 mark (c) About two months into election petition case in Ghana, a Gallup poll indicate that a near record (low) level of 41% of adults expressed "a great deal” "quite a lot” of confidence in the Ghanaian Supreme Court. Suppose that yo conducted your own telephone survey at that time and randomly called peop and asked them to describe their level of confidence in the Supreme Court. Fin the probability distribution for Y,, the number of calls until the first person found who does not express "a great deal" or "quite a lot” of confidence in ti Ghanaian Supreme Court. (5 mark

Answer 1

E(x)= ( xla+bx? 1 3 ar? bx+ + 2 = 3 +-= 2.4 2a+b= 12 5 Total probabilit y = 1 [(a+bx+)=1 0 1 ax + bx 3 = 1 0 b a+c=1 3 3a+b=3 12 a=3 5 b=3

f(x) 3 6 -+-x 0<x<1 5 5 13 6 3 6x2 +-xr-x+ 5 5 52 F(x) Ax<=0))-03 = 0.375 +

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Weighted Average À - w X1 + w, ła Wi+w2 i. 1 1 - 3w,X1+ w, X2 _371 + X2_371 X2 4w 3 4 a=-:1-a 4 ii wX1 +wX2 X1 + X2 1 X =X w+w 2 Y is y failures (to expess a great deal or quite a lot) before 1st success y = 0,1,2,3..... P(Y = y) = 0.41 (1 -0.41)=(0.59)0.41*