Assignment 3

# Assignment 3 - SMSL/07 THE UNIVERSITY OF HONG KONG...

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SMSL/07 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability and Statistics I Assignment 3 Section A 1. Let X be a discrete random variable distributed on the set { 1 , 2 , 3 , 4 , 5 , 6 } . Its mass function f ( x ) is tabulated below: x 1 2 3 4 5 6 f ( x ) 0.2 0.1 0.3 — 0.2 0.1 (a) Calculate f (4). (b) Calculate the cdf F ( x ) of X for x = 1 , 2 , 3 , 4 , 5 , 6. What is the value of F (3 . 7)? (c) Let X 1 and X 2 be two independent copies of X . Tabulate the mass function g ( x ) of X 1 + X 2 for x = 2 , 3 ,..., 12. 2. Let X be a positive random variable distributed with the density function f ( x ) = c (1 + x ) - 2 , x > 0 . (a) Calculate c . (b) Calculate P ( X x ). Hence determine the value of x such that the above probability equals 1 / 2. (c) Calculate P (1 < X < 2). 3. Let X be a continuous random variable with a continuous pdf f . (a) Find a pdf for | X | . (b) Find a pdf for X 2 . (c) Plot the pdf’s found in parts (a) and (b) for the special case where X N (0 , 1). 4. Let X be a continuous random variable with density function f ( x ) = C ( x - x 2 ) , α < x < β, C > 0. 1

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(a) What are the possible values of α and β ? (b) What is C ? 5. Let X be an exponential random variable. Show that P ( X > s + t | X > s ) = P ( X > t ) , s,t 0. 6. Let
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Assignment 3 - SMSL/07 THE UNIVERSITY OF HONG KONG...

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