Answer:
(1) -0.833
(2) 0.80
(3) 0.70
(4) 390
(5) 90
(7) 48
Step-by-step explanation:
Given:
E (X) = 100, E (Y) = 120, E (Z) = 130
Var (X) = 9, Var (Y) = 16, Var (Z) = 25
Cov (X, Y) = -10, Cov (X, Z) = 12, Cov (Y, Z) = 14
The formulas used for correlation is:
![Corr (A, B) = \frac{Cov (A, B)}{\sqrt{Var (A)\times Var(B)}} \\](https://tex.z-dn.net/?f=Corr%20%28A%2C%20B%29%20%3D%20%5Cfrac%7BCov%20%28A%2C%20B%29%7D%7B%5Csqrt%7BVar%20%28A%29%5Ctimes%20Var%28B%29%7D%7D%20%5C%5C)
(1)
Compute the value of Corr (X, Y)-
![Corr (X, Y) = \frac{Cov (X, Y)}{\sqrt{Var (X)\times Var(Y)}} \\=\frac{-10}{\sqrt{9\times16}} \\=-0.833](https://tex.z-dn.net/?f=Corr%20%28X%2C%20Y%29%20%3D%20%5Cfrac%7BCov%20%28X%2C%20Y%29%7D%7B%5Csqrt%7BVar%20%28X%29%5Ctimes%20Var%28Y%29%7D%7D%20%5C%5C%3D%5Cfrac%7B-10%7D%7B%5Csqrt%7B9%5Ctimes16%7D%7D%20%5C%5C%3D-0.833)
(2)
Compute the value of Corr (X, Z)-
![Corr (X, Z) = \frac{Cov (X, Z)}{\sqrt{Var (X)\times Var(Z)}} \\=\frac{12}{\sqrt{9\times25}} \\=0.80](https://tex.z-dn.net/?f=Corr%20%28X%2C%20Z%29%20%3D%20%5Cfrac%7BCov%20%28X%2C%20Z%29%7D%7B%5Csqrt%7BVar%20%28X%29%5Ctimes%20Var%28Z%29%7D%7D%20%5C%5C%3D%5Cfrac%7B12%7D%7B%5Csqrt%7B9%5Ctimes25%7D%7D%20%5C%5C%3D0.80)
(3)
Compute the value of Corr (Y, Z)-
![Corr (Y, Z) = \frac{Cov (Y, Z)}{\sqrt{Var (Y)\times Var(Z)}} \\=\frac{14}{\sqrt{16\times25}} \\=0.70](https://tex.z-dn.net/?f=Corr%20%28Y%2C%20Z%29%20%3D%20%5Cfrac%7BCov%20%28Y%2C%20Z%29%7D%7B%5Csqrt%7BVar%20%28Y%29%5Ctimes%20Var%28Z%29%7D%7D%20%5C%5C%3D%5Cfrac%7B14%7D%7B%5Csqrt%7B16%5Ctimes25%7D%7D%20%5C%5C%3D0.70)
(4)
Compute the value of E (3X+4Y-3Z)-
![E(3X+4Y-3Z)=3E(X)+4E(Y)-3E(Z)\\=(3\times100)+(4\times120)-(3\times130)\\=390](https://tex.z-dn.net/?f=E%283X%2B4Y-3Z%29%3D3E%28X%29%2B4E%28Y%29-3E%28Z%29%5C%5C%3D%283%5Ctimes100%29%2B%284%5Ctimes120%29-%283%5Ctimes130%29%5C%5C%3D390)
(5)
Compute the value of Var (3X-3Z)-
![Var (3X-3Z)=[(3)^{2}\times Var(X)]+[(-3)^{2}\times Var (Z)]+(2\times3\times-3\times Cov(X, Z)]\\=(9\times9)+(9\times25)-(18\times12)\\=90](https://tex.z-dn.net/?f=Var%20%283X-3Z%29%3D%5B%283%29%5E%7B2%7D%5Ctimes%20Var%28X%29%5D%2B%5B%28-3%29%5E%7B2%7D%5Ctimes%20Var%20%28Z%29%5D%2B%282%5Ctimes3%5Ctimes-3%5Ctimes%20Cov%28X%2C%20Z%29%5D%5C%5C%3D%289%5Ctimes9%29%2B%289%5Ctimes25%29-%2818%5Ctimes12%29%5C%5C%3D90)
(6)
Compute the value of Var (3X+4Y-3Z)-
![Var (3X+4Y-3Z)=[(3)^{2}\times Var(X)]+[(4)^{2}\times Var(Y)]+[(-3)^{2}\times Var (Z)]+[(2\times3\times4\times Cov(X, Y)]+[(2\times3\times-3\times Cov(X, Z)]+[(2\times4\times-3\times Cov(Y, Z)]\\=(9\times9)+(16\times16)+(9\times25)+(24\times-10)-(18\times12)-(24\times14)\\=-230](https://tex.z-dn.net/?f=Var%20%283X%2B4Y-3Z%29%3D%5B%283%29%5E%7B2%7D%5Ctimes%20Var%28X%29%5D%2B%5B%284%29%5E%7B2%7D%5Ctimes%20Var%28Y%29%5D%2B%5B%28-3%29%5E%7B2%7D%5Ctimes%20Var%20%28Z%29%5D%2B%5B%282%5Ctimes3%5Ctimes4%5Ctimes%20Cov%28X%2C%20Y%29%5D%2B%5B%282%5Ctimes3%5Ctimes-3%5Ctimes%20Cov%28X%2C%20Z%29%5D%2B%5B%282%5Ctimes4%5Ctimes-3%5Ctimes%20Cov%28Y%2C%20Z%29%5D%5C%5C%3D%289%5Ctimes9%29%2B%2816%5Ctimes16%29%2B%289%5Ctimes25%29%2B%2824%5Ctimes-10%29-%2818%5Ctimes12%29-%2824%5Ctimes14%29%5C%5C%3D-230)
But this is not possible as variance is a square of terms.
(7)
Compute the value of Cov (3X, 2Y+3Z)-
![Cov(3X, 2Y+3Z)=Cov(3X,2Y)+Cov(3X, 3Z)\\=6Cov(X, Y)+9Cov(X,Z)\\=(6\times-10)+(9\times12)\\=48](https://tex.z-dn.net/?f=Cov%283X%2C%202Y%2B3Z%29%3DCov%283X%2C2Y%29%2BCov%283X%2C%203Z%29%5C%5C%3D6Cov%28X%2C%20Y%29%2B9Cov%28X%2CZ%29%5C%5C%3D%286%5Ctimes-10%29%2B%289%5Ctimes12%29%5C%5C%3D48)