Let X1, X2 , and X3 be independent standard normal random variable. If we defined Y1=X2, Y2=X1+X2/2 and Y3=X1+ X2 +X3/3.Then find then joint pdf of Y1, Y2, and Y3 using the Jacobian method?
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Expert's answer
2022-05-12T14:43:19-0400
Suppose that random variables X1, X2, X3 have normal distributions with parameters μ1,σ12, μ2,σ22 and μ3,σ32 respectively. We use a known fact that in case a random variable Z has a normal distribution with parameters μZ,σZ2, then, a random variable aZ, where a∈R, has a normal distribution with parameters aμZ,a2σZ2. We receive, that 2X2 has a normal distribution with parameters 2μ2,4σ22 and 3X3 has a normal distribution with parameters: 3μ3,9σ32. We use also a known fact, that a sum of normally distributed variables is again a normally distributed variable. We get, that Y2 has a normal distribution with parameters: μ1+2μ2,σ12+4σ22 and Y3 has a normal distribution with parameters: μ1+μ2+3μ3,σ12+σ22+9σ32. Denote: α1=μ1+2μ2, β1=σ12+4σ22, α2=μ1+μ2+3μ3, β2=σ12+σ22+9σ32. We use a known fact that the joint probability density function is equal to the product of density functions for independent variables. We get: pY1,Y2,Y3(y1,y2,y3)=pY1(y1)pY2(y2)pY3(y3)=σ2β1β2(2π)31e−21(σ2y1−μ2)2−21(β1y2−α1)2−21(β2y3−α2)2
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