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The random variables.\\ and I have joint probability density function (pdf) fxy(x,y) = 24xBy for 0 ) together with the range of X for which it is defined. [2] (iv) Determine the conditional probability P(x > ,ly - !) [2] (v) Determine the conditional expectation E( X|Y - !). [3] (vi) Verify that ETE[X" | Y1] = ETX] by evaluating each side of the equation. 137 [Total 17] A large pension scheme regularly investigates the lifestyle of its pensioners using surveys. In successive surveys it draws a random sample from all pensioners in the scheme and it obtains the following data on whether the pensioners smoke. Survey 1: Of 124 pensioners surveyed, 36 were classed as smokers. Survey 2: Of 136 pensioners surveyed, 25 were classed as smokers. An actuary wants to investigate, using statistical testing at a $% significance level, whether there have been significant changes in the proportion of pensioners, p. who smoke in the entire pension scheme. (i) Perform a statistical test, without using a contingency table, to determine if the proportion p has changed from the first survey to the second. [7] When a third survey is performed it is found that 26 out of the 141 surveyed pensioners, are smokers. (ii) Perform a statistical test using a contingency table to determine if the proportion p is different among the three surveys. [7] (iii) (a) Calculate the proportion of smokers in the third survey. (b) Comment on your answers to parts (i) and (ii). [2] [Total 16]