Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
Chapter 2 Multivariate Distributions 2.1 DistributionsofTwoRandomVariables We begin the discussion of a pair of random variables ...
86 Multivariate Distributions with random variables in Section 1.5 we can uniquely definePX 1 ,X 2 in terms of the cumulative di ...
2.1. Distributions of Two Random Variables 87 At times it is convenient to speak of thesupportof a discrete random vec- tor (X 1 ...
88 Multivariate Distributions In the next example, we use the general fact that double integrals can be ex- pressed as iterated ...
2.1. Distributions of Two Random Variables 89 x y z Figure 2.1.1: A sketch of the the surface of the joint pdf discussed in Exam ...
90 Multivariate Distributions Table 2.1.1: Joint and Marginal Distributions for the discrete random vector (X 1 ,X 2 ) of Exampl ...
2.1. Distributions of Two Random Variables 91 x 2 x 1 12 p 1 (x 1 ) 1 101 101 102 2 101 102 103 3 102 103 105 p 2 (x 2 ) 104 106 ...
92 Multivariate Distributions −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 x y ( x, − 1 −x^2 ) ( x, 1 −x^2 ) Region of Integratio ...
2.1. Distributions of Two Random Variables 93 (0,1). The reader should sketch this region on the space of (X 1 ,X 2 ). Fixingx 1 ...
94 Multivariate Distributions Likewise if (X 1 ,X 2 ) is discrete, thenE(Y)existsif ∑ x 1 ∑ x 2 |g(x 1 ,x 2 )|pX 1 ,X 2 (x 1 ,x ...
2.1. Distributions of Two Random Variables 95 Example 2.1.8.LetX 1 andX 2 have the pdf f(x 1 ,x 2 )= { 8 x 1 x 2 0 <x 1 <x ...
96 Multivariate Distributions 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x y ( 0 , y )( y, y ) Figure 2.1.3: Region of inte ...
2.1. Distributions of Two Random Variables 97 Example 2.1.10.Let the continuous-type random variablesXandY have the joint pdf f( ...
98 Multivariate Distributions 2.1.2.LetA 1 ={(x, y):x≤ 2 ,y≤ 4 },A 2 ={(x, y):x≤ 2 ,y≤ 1 },A 3 = {(x, y):x≤ 0 ,y≤ 4 },andA 4 ={( ...
2.1. Distributions of Two Random Variables 99 2.1.8.LetXandYhave the pdff(x, y)=1, 0 <x< 1 , 0 <y<1, zero elsewhere. ...
100 Multivariate Distributions 2.2 Transformations:BivariateRandomVariables............. Let (X 1 ,X 2 ) be a random vector. Sup ...
2.2. Transformations: Bivariate Random Variables 101 A random variable of interest isY 1 =X 1 +X 2 ; i.e., the total number of r ...
102 Multivariate Distributions SinceFZ′(z) exists for all values ofz, the pmf ofZmay then be written fZ(z)= ⎧ ⎨ ⎩ z 0 <z< ...
2.2. Transformations: Bivariate Random Variables 103 SinceBis arbitrary, the last integrand must be the joint pdf of (Y 1 ,Y 2 ) ...
104 Multivariate Distributions x 2 x 1 = 0x 1 = 1 x 2 = 1 (0, 0) x 2 = 0 x 1 S Figure 2.2.2:The support of (X 1 ,X 2 ) of Exampl ...
«
1
2
3
4
5
6
7
8
9
10
»
Free download pdf