{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture 10 - Expectation of a Random Variable

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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern