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Unformatted text preview: e numerical value of P {(3X − 2Y )2 ≤ 28}. (c) Find the numerical value of E [Y |X = 3]. Solution: (a) Var(3X − 2Y ) = Cov(3X − 2Y, 3X − 2Y ) = 9Var(X ) − 12Cov(X, Y ) + 4Var(Y ) = 9 − 6 + 4 = 7. (b) Also, the random variable 3X − 2Y has mean zero and is Gaussian. Therefore, √ √ − P {(3X − 2Y )2 ≤ 28} = P {− 28 ≤ 3X − 2Y ≤ 28} = 2P 0 ≤ 3X√72Y ≤ 278 = 2(Φ (2) − 0.5) ≈ 0.9545. (c) Since X and Y are jointly Gaussian, E [Y |X = 3] = L∗ (3), so plugging numbers into (4.33) yields: 3 − µX E [Y |X = 3] = µY + σY ρX,Y = 3ρX,Y = 1.5. σX Chapter 5 Wrap-up The topics in these notes are listed in both the table of contents and the index. This chapter briefly summarizes the material, while highlighting some of the connections among the topics. The probability axioms allow for a mathematical basis for modeling real-world systems with uncertainty, and allow for both discrete-type and continuous-type random variables. Counting problems naturally arise for calculating probabilities when all outcomes are equally likely, and a recurring idea for counting the total number of ways something can be done is to do it sequentially, such as in, “for each choice of the first ball, there are n2 ways to choose the second ball,” and so on. Working with basic probabilites includes working with Karnaugh maps, de Morgan’s laws, and definitions of conditional probabilities and mutual independence of two or more events. Our intuition can be s...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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