stat1301test_makeup - 3. [10 marks] Let X be a random...

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09/10 p. 1 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability and Statistics I Class Test (Makeup) A n s w e r a l l q u e s t i o n s . T i m e a l l o w e d : 5 0 m i n s Name ______________________ Student ID _________________ Class ___________ 1. [10 marks] (a) State the definition of mutually exclusive events . (b) State the definition of independent events . 2. [10 marks] There are two identical coins except that one is fair with an equal probability of landing on the head and tail if it is tossed, while the other is biased with a probability of 0.7 to land on the head. In order to determine which coin is fair, we randomly selected a coin and tossed it for 10 times. Suppose that we observed 6 heads, then what is the probability that the selected coin is fair?
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Unformatted text preview: 3. [10 marks] Let X be a random variable distributed as ( ) 16 , 7 N . (a) Suppose ⎣ ⎦ X Y = (the integer part of X ). Calculate ( ) 8 = Y P . (b) Let . Find the r th moment of W . . [10 marks] lifetime characterizes the remaining survival time of a subject, given that the X e W = 4 The residual subject has already survived up to time t . Correspondingly, the mean residual lifetime is the remaining life expectancy conditional on survival at time t , which is defined as ( ) ( ) t T t T E t m > − = | for where T is the failure time. Show that the mean residual lifetime has an explicit one-to-one ≥ t correspondence to the survival function ( ) ( ) ( ) ( ) ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − = ∫ t du u m t m m t S 1 exp ....
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This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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