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Assignment 4 - 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 4 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 0.1 0.2 0.1 (a) Calculate E [ X ]. (b) Calculate E [ X 2 ]. (c) Calculate Var( X ). 2. Let X be a positive random variable distributed with the density function f ( x ) = c (1 + x ) - 3 , x > 0 . (a) Calculate c . (b) Calculate the mean of X . (c) Calculate the median of X . (d) Calculate the interquartile range of X . 3. Princess Aurora (better known as the Sleeping Beauty ) falls asleep under the spell of Fairy Carabosse. All it takes to awaken Aurora is a magical kiss of a prince. A group of N princes are prepared to offer their kisses. Among them is Prince D´ esir´ e, the only person whose kiss can awaken Aurora. Princes are to be selected at random to kiss Aurora successively until she is awakened. The court decides to reward each selected prince with $1 for his kiss, whether or not it is successful, and considers the following two options for the selection procedure: Option 1. Any unsuccessful princes are kept entertained in a luxurious palace, on which the court has to spend $1 per prince for entertainment, thus preventing the unsuccessful princes from queuing up again to kiss Aurora and claim the $1 rewards repeatedly.
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