Q7 - one of his outings is 40 minutes with a standard...

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EXERCISES IN STATISTICS Series A, No. 7 1. Let x and y be jointly distributed random variables with conditional ex- pectations which can be written as E ( y | x ) = α + βx and E ( x | y ) = + δy . Express β and δ in terms of the moments of the joint distributions and show that β 1 . 2. A marksman’s scores are a sequence of random variables y i with E ( y i ) = 90 and V ( y i ) = 16 for all i . The correlation between successive scores is 0.9, and the expectation of a score conditional upon the previous score is given by E ( y i | y i 1 ) = a + by i 1 where a = (1 b ) E ( y i ). Find the expected score given that the previous score was 80. 3. The expected rainfalls in September, October and November are 10 ins, 8 ins 6 ins respectively, with a variance–covariance matrix of 6 3 1 . 5 3 6 3 1 . 5 3 6 . Calculate the expected rainfall throughout these three months and ﬁnd its variance. If the September rain was unusually high, in what direction would ones estimates of rainfall be revised (a) in the two months following, and (b) for the three month period? 4. A man runs on Hampstead Heath twice a week. The average duration of
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Unformatted text preview: one of his outings is 40 minutes with a standard deviation of 5 minutes. His average running time per week is 1 hour 20 minutes with a standard deviation of 4 minutes. Given that he ran for only 15 minutes on Monday, what is the expected duration of his Friday outing? 5. An investor has a choice of three ﬁnancial assets. The expected yields of these assets are given in the vector [ 0 . 04 . 03 . 05 ] and the variances and covariances of the yields are given in the matrix 10 − 4 3 . 29 − . 83 − . 83 3 . 41 2 . 01 . Derive expressions for the expected yield and variance of a portfolio con-taining ∏Q of the ﬁrst asset and (1 − ∏ ) Q of the second asset, and ascertain whether there is any value of ∏ such that the variance of this portfolio is less than the variance of an investment Q in the third asset....
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