Re: [商管] [統計]-高應大97-金資所

[solaarr ()]

※ 引述《aiaorry (我不想皺眉頭)》之銘言：
: If X and Y are joint normal with μx=1, μy= -1, σx=2, σy=1 and correlation
: ρ= -0.7,
: 1. Calculate fx(x) and fy(y), the marginal p.d.f. of random variables X and Y
:    respectively.

(X,Y)~BVN(μx=1, μy= -1, σx=2, σy=1, ρ= -0.7)

              2
則X~N(μx, σx )
              2
  Y~N(μy, σy )


: 2. Evaluate the conditonal mean and conditonal variance of Y given X=x.
            σy              2     2
Y|x~N(μy+ρ____(X-μx) , σy (1-ρ ))
            σx

: 3. If Z1 = X+Y , Z2 = 3X-Y. Evaluate the correlation between Z1 and Z2.

Cov(Z1,Z2)=Cov(X+Y,3X-Y)
          =3Cov(X,X)-Cov(X,Y)+3Cov(X,Y)-Cov(Y,Y)
          =3Var(X)+2Cov(X,Y)-Var(Y)
               2                2
          =3σx   +2ρσxσy-σy

          =12-2.8-1
          =8.2

Var(Z1)= Var(X+Y)
       = Var(X)+Var(Y)+2Cov(X,Y)
       = 4+1-2.8
       = 2.2

Var(Z2)= Var(3X-Y)
       = 9Var(X)+Var(Y)+2*3*(-1)*Cov(X,Y)
       = 36+1-6*(-1.4)
       = 45.4

correlation= Cov(Z1,Z2)/根號Var(Z1)Var(Z2)
           = 0.8205


(前兩小題把數字代進去即可@@~)



: 這種題目的邊際分配是不是就不獨立了呢?!
: 該怎麼計算呢?!
: 謝謝

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