A2-431F08 - ASSIGNMENT 2 ACTSC 431/831 FALL 2008 Due at the...

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ASSIGNMENT 2 – ACTSC 431/831, FALL 2008 Due at the beginning of the tutorial on Wednesday, October 8 . 1. Let two independent random variables X 1 and X 2 have distributions F 1 and F 2 , re- spectively. Actuary A assumes that the loss of a portfolio is Y A and the distribu- tion of Y A is the average of the distributions F 1 and F 2 , namely, the cdf of Y A is F Y A ( y ) = 1 2 ( F 1 ( y ) + F 2 ( y )) for y ( -∞ , ), while actuary B assumes the loss of the same portfolio is Y B , which is the average of X 1 and X 2 , namely Y B = 1 2 ( X 1 + X 2 ). (a) Assume that F 1 is a normal distribution N (10 , 5) and F 2 is a normal distribution N (50 , 8). i. Calculate the means of Y A and Y B . [2 marks] ii. Calculate the variances of Y A and Y B . [4 marks] iii. Prove or disprove that Y A has a normal distribution. [8 marks] iv. Prove or disprove that Y B has a normal distribution. [6 marks] (b) Assume that F 1 is an exponential distribution with mean 10 and F 2 is an expo- nential distribution with mean 50.
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