Introduction to Probability and Statistics for Engineers and Scientists
104 Chapter 4:Random Variables and Expectation In particular, for anynsets of real numbersA 1 ,A 2 ,...,An P{X 1 ∈A 1 ,X 2 ∈A 2 ...
4.3Jointly Distributed Random Variables 105 *4.3.2 Conditional Distributions The relationship between two random variables can o ...
106 Chapter 4:Random Variables and Expectation EXAMPLE 4.3g Suppose thatp(x,y), the joint probability mass function ofXandY,is g ...
4.4Expectation 107 EXAMPLE 4.3h The joint density ofXandYis given by f(x,y)= { 12 5 x(2−x−y)0<x<1, 0<y<^1 0 otherwis ...
108 Chapter 4:Random Variables and Expectation then E[X]= 0 ( 1 3 ) + 1 ( 2 3 ) =^23 is a weighted average of the two possible v ...
4.4Expectation 109 repetitions of the experiment. That is, if we continually roll a fair die, then after a large number of rolls ...
110 Chapter 4:Random Variables and Expectation However, if we define the functionGby G(p)=I(2−p) then we see from the above that ...
4.5Properties of the Expected Value 111 -1 p (-1) = .10, center of gravity = .9 0 p (0) = .25, 1 p (1) = .30, 2 p (2) = .35 FIGU ...
112 Chapter 4:Random Variables and Expectation value ofX, but the expected value of some function ofX, sayg(X). How do we go abo ...
4.5Properties of the Expected Value 113 By differentiatingFY(a), we obtain the density ofY, fY(a)= 1 3 a−2/3,0≤a< 1 Hence, E[ ...
114 Chapter 4:Random Variables and Expectation EXAMPLE 4.5c Applying Proposition 4.5.1 to Example 4.5a yields E[X^2 ]= 02 (0.2)+ ...
4.5Properties of the Expected Value 115 That is, the expected value of a constant is just its value. (Is this intuitive?) Also, ...
116 Chapter 4:Random Variables and Expectation For instance, E[X+Y+Z]=E[(X+Y)+Z] =E[X+Y]+E[Z] by Equation 4.5.1 =E[X]+E[Y]+E[Z] ...
4.5Properties of the Expected Value 117 where Xi= { 1 if theith letter is placed in its proper envelope 0 otherwise Now, since t ...
118 Chapter 4:Random Variables and Expectation when the last equality follows since each of the 10 coupons will (independently) ...
4.6Variance 119 Because we expectXto take on values around its meanE[X], it would appear that a reasonable way of measuring the ...
120 Chapter 4:Random Variables and Expectation Hence, since it was shown in Example 4.4a thatE[X]=^72 , we obtain from Equation ...
4.7Covariance and Variance of Sums of Random Variables 121 That is, the variance of a constant plus a random variable is equal t ...
122 Chapter 4:Random Variables and Expectation =E[XY]−μxμy−μyμx+μxμy =E[XY]−E[X]E[Y] (4.7.1) From its definition we see that cov ...
4.7Covariance and Variance of Sums of Random Variables 123 Proof Cov ∑n i= 1 Xi, ∑m j= 1 Yj = ∑n i= 1 Cov Xi, ∑m j= 1 ...
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