Ch3 Mathematical Expectation

Ch3 Mathematical Expectation - Stat1801 Probability and...

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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Chapter III Mathematical Expectation § 3.1 Expectation Example 3.1 Consider the following two games. In each game, three fair dice will be rolled. Game 1: If all the dice face up with same number, then you win $24 otherwise you lose $1. Game 2: You win $1, $2, or $3, according to one die face up as six, two dice face up as six or three dice face up as six, respectively. If no dice face up as six, then you lose $1. Which game is a better choice? To make a better decision, one may consider the amount one will win (or lose) in the long run. First we need to evaluate the probabilities of win or lose in each game. Let X, Y be the amounts of money you will win in one single trial of game 1 and game 2 respectively. A negative value means you lose money. For game 1, () ( ) 36 1 6 1 6 1 6 1 6 dice three on number same 24 = × × × = = = P X P 36 35 36 1 1 1 = = = X P For game 2, 72 25 6 5 6 5 6 1 3 1 = × × × = = Y P , 72 5 6 1 6 1 6 5 3 2 = × × × = = Y P 216 1 6 1 6 1 6 1 3 = × × = = Y P , 216 125 216 1 72 5 72 25 1 1 = = = Y P Suppose we play game 1 for 36000 times. Since the relative frequency is a good estimate of the probability when number of trials is large, approximately in 1000 times we will win $24 and 35000 times we will lose $1. So in these 36000 trials of game 1, we win ( ) 11000 35000 1 1000 24 = × + × Approximately we will lose $11000 in 36000 trial of game 1. The average amount we win in each trial is p. 61
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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 36 11 36000 11000 = This is the long term average of gain if we play game 1 repeatedly. Indeed it can be calculated as ( ) 36000 35000 1 1000 24 36 11 × + × = () 36 35 1 36 1 24 × + × = ( ) ( ) 1 1 24 24 = × + = × = X P X P Similarly, the long term average of gain if we play game 2 repeatedly is ( ) ( ) ( ) ( ) 1 1 3 3 2 2 1 1 = × + = × + = × + = × Y P Y P Y P Y P 36 11 216 17 216 125 1 216 1 3 72 5 2 72 25 1 > = × + × + × + × = Therefore game 2 is better than game 1 in terms of long term average gain. However, since in the long run you will lose money in both game, the best strategy is do not gamble at all. Definition Let X be a random variable with pmf ( ) x p or pdf ( ) x f . The mathematical expectation ( expected value ) of X is defined by () ( Ω = X x x xp X E ) (discrete) or ( ) ( ) = dx x xf X E (continuous) provided that the summation/integral exists. In general, for any function g , the expected value of ( ) X g is ( ) ( ) Ω = X x x p x g X g E or ( ) ()() = dx x f x g X E . e.g. ( ) () Ω = X x x p x X E 2 2 , ( ) ( ) = log log dx x f x X E , …, etc. p. 62
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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Example 3.2 Suppose the waiting time T for the occurrence of a certain event is distributed according to the pdf () 10 10 1 t e t f = , < < t 0. The expected waiting time is 10 10 1 0 10 = = dt te T E t time units.
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Ch3 Mathematical Expectation - Stat1801 Probability and...

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