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Unformatted text preview: APPENDIX B Fundamentals of probability (cont.) 1. EXPECTED VALUE The probability distribution of every RV has 2 main features: • Measures of central tendency: median, expected value , mean • Measures of variability or spread: variance and standard deviation If X is a discrete RV, with k possible values, say, { x 1 ,x 2 ,…x k }, then the expected value is given by: ? = = ¡ ?(¡ ) ¢1 If X is a continuous RV, then the expected value is given by: ? = £ ¤ ¡? ¡ ?¡ ∞ −∞ The expected value is also called the population mean Suppose X is a discrete RV and we define a new RV g(X) for any function g, then the expected value of g(X) is given by: E g X = ?(¡ )? ¥ (¡ ) ¢1 If X is a continuous RV, E g X = ¤ g x ? ¥ ∞ −∞ ¡ ?¡ 1. EXPECTED VALUE: an example Random variable X= x i Probability f(x i ) = p i = P(X= x i ) pdf x 1 = 1 1/8 x 2 = 0 1/2 x 3 = 2 3/8 ? = 1 ¡¢ The expected value of X is _____ The expected value of X 2 is _____ 1. EXPECTED VALUE: an example Random variable X= x i Probability f(x i ) = p i = P(X= x i ) pdf x 1 = 1 1/8 x 2 = 0 1/2 x 3 = 2 3/8 ? = 1 ¡¢ The expected value of X is _5/8____ The expected value of X 2 is __13/8___ 1. EXPECTED VALUE: properties Suppose we have a new RV that is a linear function of X. e.g. Y= a + bX What is the expected value of Y? • (Long way) we compute the pdf of Y • (Easy way) use properties of expected value 1. EXPECTED VALUE: properties Property 1: For any constant c, (?) = ? Property 2: For any constant c, (??) = ?(?) Property 3: For any constant c, ? + ? = ? + (?) The above properties imply : For any constants a and b, ? + ?? = ? + ?(?) Property 4: If a 1 . a 2 , … a k are constants and X 1 , X 2 , … X k are random variables, then: ?...
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This note was uploaded on 09/25/2011 for the course ECON 322 taught by Professor Francisco during the Spring '07 term at Rutgers.
 Spring '07
 Francisco

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