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exam_05_final - MT1141 Probability and Statistics I exam...

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MT1141 Probability and Statistics I exam Rubric 2 hour exam Section A: Answer all four questions. Section B: Answer two questions from three. Sections A and B carry equal weighting. The exam is marked out of 120. To be provided: Mathematical Formula Tables; Normal distribution tables and the t-distribution tables. 1
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MT1141 Section A A. 1 a) For two sets A and B , prove that [6] A C B C = ( A B ) C . b) State the mathematical definition of independence for two events A and B . [2] Given that A and B are independent. c) Show that A C and B C are independent. (Hint: Use part a ). [3] d) Show that P ( A B | A ) = P ( B ). [3] A. 2 The probability mass function of the random variable Y is given in the table below. y 0 1 2 3 4 P ( Y = y ) α 1 6 1 4 β 1 3 a) Given that E [ Y ] = 5 2 . Show that α = 1 12 and β = 1 6 . [6] b) Calculate E [ Y 2 ], and hence, or otherwise, calculate var (2 Y + 3). [6] c) Given that E [ Y 3 ] = 28, calculate E [( Y + 1) 3 ]. [3] A. 3 The probability density function of a continuous random variable, X , is given by f ( x ) = cx 2 0 x < 1 c 1 x < 2 c (3 - x ) 2 x < 3 0 otherwise. a) Show that c = 6 11 . [6] b) Calculate the cumulative distribution function F ( x ) of the random variable X. [9] 2
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A. 4 A soft drinks company produces 330 ml bottles of cola. The volume of cola in each bottle is approximately normally distributed with mean 330 ml and standard deviation 8 ml .
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