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lecture3_complete - APPENDIX B Fundamentals of...

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APPENDIX B Fundamentals of probability (cont.)
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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: 𝐸 ? = 𝜇 = ? 𝑖 ?(? 𝑖 ) 𝑘 𝑖=1 If X is a continuous RV, then the expected value is given by: 𝐸 ? = ?? ? ?? −∞ 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 = ?(? 𝑖 )? ? (? 𝑖 ) 𝑘 𝐼=1 If X is a continuous RV, E g X = g x ? ? −∞ ? ??
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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 ? 𝑖 = 1 𝑛 𝑖=1 The expected value of X is _____ The expected value of X 2 is _____
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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 ? 𝑖 = 1 𝑛 𝑖=1 The expected value of X is _5/8____ The expected value of X 2 is __13/8___
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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
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1. EXPECTED VALUE: properties Property 1: For any constant c, 𝐸(?) = ? Property 2: For any constant c, 𝐸(??) = ?𝐸(?) Property 3: For any constant c, 𝐸 ? + ? = ? + 𝐸(?) The above properties imply : For any constants a and b, 𝐸 ? + ?? = ? + ?𝐸(?) Property 4: If a 1 . a 2 , … a k are constants and X 1 , X 2 , … X k are random variables, then: 𝐸 ?
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