Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
2.7. Transformations for Several Random Variables 145 which reduces to the supportT ofY 1 ,Y 2 ,Y 3 of T ={(y 1 ,y 2 ,y 3 ): 0&l ...
146 Multivariate Distributions Hence the joint pdf ofY 1 ,Y 2 ,Y 3 is g(y 1 ,y 2 ,y 3 )=y 32 e−y^3 , (y 1 ,y 2 ,y 3 )∈T. The mar ...
2.7. Transformations for Several Random Variables 147 f(0) = 0? Then our newSisS={−∞<x<∞butx =0}.Wethentake A 1 ={x:−∞< ...
148 Multivariate Distributions may not be one-to-one. Suppose, however, that we can representSas the union of a finite number, s ...
2.7. Transformations for Several Random Variables 149 The value of the first Jacobian is J 1 = ∣ ∣ ∣ ∣ ∣ ∣ 1 2 √ y 2 /y (^112) √ ...
150 Multivariate Distributions This, however, is the mgf of the pmf pY(y)= { (μ 1 +μ 2 +μ 3 )ye−(μ^1 +μ^2 +μ^3 ) y! y=0,^1 ,^2 . ...
2.8. Linear Combinations of Random Variables 151 2.7.5.LetX 1 ,X 2 ,X 3 be iid with common pdff(x)=e−x,x>0, 0 elsewhere. Find ...
152 Multivariate Distributions Proof:Using the definition of the covariance and Theorem 2.8.1, we have the first equality below, ...
2.8. Linear Combinations of Random Variables 153 Example 2.8.2(Sample Variance).Define thesample varianceby S^2 =(n−1)−^1 ∑n i=1 ...
154 Multivariate Distributions 2.8.9.Letμandσ^2 denote the mean and variance of the random variableX.Let Y=c+bX,wherebandcare re ...
Chapter 3 Some Special Distributions 3.1 TheBinomialandRelatedDistributions................ In Chapter 1 we introduced theunifor ...
156 Some Special Distributions ith trial. An observed sequence ofnBernoulli trials is then ann-tuple of zeros and ones. In such ...
3.1. The Binomial and Related Distributions 157 which is not a simple calculation. Most statistical packages provide procedures ...
158 Some Special Distributions Example 3.1.3.IfY isb(n,^13 ), thenP(Y ≥1) = 1−P(Y =0)=1−(^23 )n. Suppose that we wish to find th ...
3.1. The Binomial and Related Distributions 159 Theorem 3.1.1.LetX 1 ,X 2 ,...,Xmbe independent random variables such that Xihas ...
160 Some Special Distributions Ifr=1,thenYhas the pmf pY(y)=p(1−p)y,y=0, 1 , 2 ,..., (3.1.4) zero elsewhere, and the mgfM(t)=p[1 ...
3.1. The Binomial and Related Distributions 161 where the multiple sum is taken over all nonnegative integers and such thatx 1 + ...
162 Some Special Distributions 3.1.3 HypergeometricDistribution In Chapter 1, for a particular problem, we introduced the hyperg ...
3.1. The Binomial and Related Distributions 163 EXERCISES 3.1.1.If the mgf of a random variableXis (^13 +^23 et)^5 , findP(X= 2 ...
164 Some Special Distributions x<-0:15; par(mfrow=c(3,3)); p <- 1:9/10 for(j in p){plot(dbinom(x,15,j)~x); title(paste("p= ...
«
4
5
6
7
8
9
10
11
12
13
»
Free download pdf