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A03ans - EXERCISES IN STATISTICS Series A No 3 0 x 1...

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EXERCISES IN STATISTICS Series A, No. 3 1. If f ( x ) = a + bx 2 ; 0 x 1, determine a and b such that E ( x ) = 3 4 . Answer: The condition that f ( x ) dx = 1, which is fulfilled by any probability density function, can be expressed as 1 0 ( a + bx 2 ) dx = ax + bx 3 3 1 0 = 1 . (1) The condition that E ( x ) = 3 / 4 can be written as 1 0 x ( a + bx 2 ) dx = ax 2 2 + bx 4 4 1 0 = 3 4 . (2) From (1) we have a + b 3 = 1 ⇐⇒ 3 a + b = 3 , and, from (2), we have a 2 + b 4 = 3 4 ⇐⇒ 2 a + b = 3 . It follows that a = 0, b = 3. 2. Let f ( x ) = 1; 0 x 1. Find (a) the mean and variance of x , (b) the mean and variance of x 2 . Answer: E ( x ) = 1 0 xdx = x 2 2 1 0 = 1 2 E ( x 2 ) = 1 0 x 2 dx = x 3 3 1 0 = 1 3 V ( x ) = E ( x 2 ) − { E ( x ) } 2 = 1 3 1 4 = 1 12 . Next E ( x 4 ) = 1 0 x 4 dx = x 5 5 1 0 = 1 5 , 1
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SERIES A No.3 : ANSWERS V ( x 2 ) = E ( x 4 ) E ( x 2 ) 2 = 1 5 1 9 = 4 45 . 3. The probability that x buses will pass me before the 106 arrives is given by f ( x ) = 1 4 ( 3 4 ) x . What is the probability that another five buses will pass me before the 106 arrives, given that three have already passed? Answer: The buses which stop here are the 106 the 253 and the 277. We have P (106) = 1 4 and P (277 254) = 1 P (106) = 3 4 as the probabilities for the next arrival. Moreover, these probabilities are independent. Therefore the
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