Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
2.2. Transformations: Bivariate Random Variables 105 y 2 (0, 0) y 1 y 2 = y 1 y 2 = 2 – y 1 y 2 = – y 1 y 2 = y 1 – 2 T Figure 2 ...
106 Multivariate Distributions hence the joint pdf ofY 1 andY 2 is fY 1 ,Y 2 (y 1 ,y 2 )= { | 2 | 4 e −y 1 −y (^2) (y 1 ,y 2 )∈T ...
2.2. Transformations: Bivariate Random Variables 107 In addition to the change-of-variable and cdf techniques for finding distri ...
108 Multivariate Distributions So the mgf ofY=(1/2)(X 1 −X 2 )isgivenby E(etY)= ∫∞ 0 ∫∞ 0 et(x^1 −x^2 )/^2 1 4 e−(x^1 +x^2 )/^2 ...
2.3. Conditional Distributions and Expectations 109 2.2.6.SupposeX 1 andX 2 have the joint pdffX 1 ,X 2 (x 1 ,x 2 )=e−(x^1 +x^2 ...
110 Multivariate Distributions For any fixedx 1 withpX 1 (x 1 )>0, this functionpX 2 |X 1 (x 2 |x 1 ) satisfies the con- diti ...
2.3. Conditional Distributions and Expectations 111 We often abbreviate these conditional pdfs byf 1 | 2 (x 1 |x 2 )andf 2 | 1 ( ...
112 Multivariate Distributions and f 2 (x 2 )= {∫x 2 0 2 dx^1 =2x^20 <x^2 <^1 0elsewhere. The conditional pdf ofX 1 ,given ...
2.3. Conditional Distributions and Expectations 113 zero elsewhere. The conditional pdf ofX 2 ,givenX 1 =x 1 ,is f 2 | 1 (x 2 |x ...
114 Multivariate Distributions Theorem 2.3.1.Let(X 1 ,X 2 )be a random vector such that the variance ofX 2 is finite. Then, (a)E ...
2.3. Conditional Distributions and Expectations 115 which completes the proof. Intuitively, this result could have this useful i ...
116 Multivariate Distributions 2.3.2.Letf 1 | 2 (x 1 |x 2 )=c 1 x 1 /x^22 , 0 <x 1 <x 2 , 0 <x 2 <1, zero elsewhere, ...
2.4. Independent Random Variables 117 2.3.9.Five cards are drawn at random and without replacement from an ordinary deck of card ...
118 Multivariate Distributions Suppose that we have an instance wheref 2 | 1 (x 2 |x 1 ) does not depend uponx 1 .Then the margi ...
2.4. Independent Random Variables 119 SinceX+Y≤4, it would seem thatXandYare dependent. To see that this is true by definition, ...
120 Multivariate Distributions and only iff(x 1 ,x 2 )can be written as a product of a nonnegative function ofx 1 and a nonnegat ...
2.4. Independent Random Variables 121 Instead of working with pdfs (or pmfs) we could have presented independence in terms of cu ...
122 Multivariate Distributions Example 2.4.4(Example 2.4.2, Continued).Independence is necessary for condi- tion (2.4.2). For ex ...
2.4. Independent Random Variables 123 Proof.IfX 1 andX 2 are independent, then M(t 1 ,t 2 )=E(et^1 X^1 +t^2 X^2 ) = E(et^1 X^1 e ...
124 Multivariate Distributions Example 2.4.5(Example 2.1.10, Continued). Let (X, Y) be a pair of random variables with the joint ...
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