Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.5. The Correlation Coefficient 125 2.4.6.Iff(x 1 ,x 2 )=e−x^1 −x^2 , 0 <x 1 <∞, 0 <x 2 <∞, zero elsewhere, is the ...

126 Multivariate Distributions It follows by the linearity of expectation, Theorem 2.1.1, that the covariance of XandY can also ...

2.5. The Correlation Coefficient 127 and σ^21 =E(X^2 )−μ^21 = ∫ 1 0 ∫ 1 0 x^2 (x+y)dxdy− ( 7 12 ) 2 = 11 144 . Similarly, μ 2 =E ...

128 Multivariate Distributions Although the converse of Theorem 2.5.2 is not true, the contrapositive is; i.e., ifρ =0thenX andY ...

2.5. The Correlation Coefficient 129 by changing integrals to sums. LetE(Y|x)=a+bx.From E(Y|x)= ∫∞ −∞ yf(x, y)dy f 1 (x) =a+bx, ...

130 Multivariate Distributions which is the desired result. Note that if the variance, Equation (2.5.9), is denoted byk(x), then ...

2.5. The Correlation Coefficient 131 Example 2.5.5.To illustrate how the correlation coefficient measures the intensity of the c ...

132 Multivariate Distributions For instance, in a simplified notation that appears to be clear, μ 1 = E(X)= ∂M(0,0) ∂t 1 μ 2 = E ...

2.5. The Correlation Coefficient 133 In each case compute the correlation coefficient ofXandY. 2.5.2.LetXandYhave the joint pmf ...

134 Multivariate Distributions (x 1 ,x 2 ) (0,0) (0,1) (0,2) (1,1) (1,2) (2,2) p(x 1 ,x 2 ) 121 122 121 123 124 121 Findp 1 (x 1 ...

2.6. Extension to Several Random Variables 135 and (b) its integral over all real values of its argument(s) is 1. Likewise, a po ...

136 Multivariate Distributions Yj=uj(X 1 ,...,Xn)forj=1,...,mand eachE(Yi) exists, then E ⎡ ⎣ ∑m j=1 kjYj ⎤ ⎦= ∑m j=1 kjE[Yj], ( ...

2.6. Extension to Several Random Variables 137 pdf of the particular group ofkvariables, provided that the latter pdf is positiv ...

138 Multivariate Distributions for independent random variablesX 1 andX 2 becomes, for mutually independent random variablesX 1 ...

2.6. Extension to Several Random Variables 139 Example 2.6.2.LetX 1 ,X 2 ,andX 3 be three mutually independent random vari- able ...

140 Multivariate Distributions Obviously, ifi =j,wehave pij(xi,xj)≡pi(xi)pj(xj), and thusXiandXjare independent. However, p(x 1 ...

2.6. Extension to Several Random Variables 141 Then E[A 1 W 1 +A 2 W 2 ]=A 1 E[W 1 ]+A 2 E[W 2 ] (2.6.11) E[A 1 W 1 B]=A 1 E[W 1 ...

142 Multivariate Distributions Proof: Use Theorem 2.6.2 to derive (2.6.15); i.e., Cov(X)=E[(X−μ)(X−μ)′] = E[XX′−μX′−Xμ′+μμ′] = E ...

2.7. Transformations for Several Random Variables 143 2.6.4.A fair die is cast at random three independent times. Let the random ...

144 Multivariate Distributions together with the inverse functions x 1 =w 1 (y 1 ,y 2 ,...,yn),x 2 =w 2 (y 1 ,y 2 ,...,yn),...,x ...