Lecture 10 - Expectation of a Random Variable

Lecture 10 - Expectation of a Random Variable - 1...

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1 EXPECTATION Lecture 10 ORIE3500/5500 Summer2011 Li 1 Expectation 1.1 Expectation of a Random Variable Expectation or expected value of a random variable denoted by E ( X ) of just EX is the weighted average of the values it takes, where the weight are the chances of the random variable taking that value. As the name suggests, if we were to expect some value from a random variable to take, then this would be the one. The expected value of a discrete random variable X , with pmf p X , is defined as E ( X ) = X i x i p X ( x i ) . For a continuous random variable X with density f X , the expectation is defined as E ( X ) = Z -∞ xf X ( x ) dx. Example A contestant on a quiz show is presented with two questions, questions 1 and 2, which he is to attempt to answer in some order he chooses. If he decides to try question i first, then he will be allowed to go on to question j , only if his answer to question i is correct. If his initial answer is incorrect, he is not allowed to answer the other question. The contestant is to receive V i dollars if he answers question i correctly. Suppose the probability that he knows the answer to question i is p i . Which question should he attempt to answer first? Solution: If he attempts to answer question 1 first, then he will win 0 with probability 1 - p 1 , V 1 with probability p 1 (1 - p 2 ) and V 1 + V 2 with prob- ability p 1 p 2 . Hence the expected winnings will be V 1 p 1 (1 - p 2 )+( V 1 + V 2 ) p 1 p 2 . If he attempts to answer question 2 first, then the expected winnings will be V 2 p 2 (1 - p 1 ) + ( V 1 + V 2 ) p 1 p 2 . Therefore, he should answer question 1 first if V 1 p 1 (1 - p 2 ) V 2 p 2 (1 - p 1 ), or equivalently, if V 1 p 1 1 - p 1 V 2 p 2 1 - p 2 . For example, if he is 60% certain of answering 1
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1.1 Expectation of a Random Variable 1 EXPECTATION question 1, worth 200, correctly and he is 80% certain of answering question 2, worth 100, correctly, then he should attempt to answer question 2 first, because 100 * 0 . 8 / 0 . 2 > 200 * 0 . 6 / 0 . 4 Sometimes it is difficult to compute the expectation of a random vari- able by following the definition. There are alternate approaches to compute expectation. The next example deals with one such case. Example
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Lecture 10 - Expectation of a Random Variable - 1...

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