Question 1174427: Let Z1, Z2 be RV' with correlation coefficient 1/3 between them. Further, let expectation of Z1, Z2 be 1 and -1, and variances 1, and 4, respectively. We define X := Z1 - Z2 + 2 and Y := 4Z1 + Z2 - 2. Find covariance and correlation coefficient of X and Y. (Just give the covariance, a number, and a correlation coefficient, a fraction in the form a/b separated by a comma.)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let $Z_1$ and $Z_2$ be random variables with:
* $E(Z_1) = 1$
* $E(Z_2) = -1$
* $Var(Z_1) = 1$
* $Var(Z_2) = 4$
* $Corr(Z_1, Z_2) = \frac{1}{3}$
We define:
* $X = Z_1 - Z_2 + 2$
* $Y = 4Z_1 + Z_2 - 2$
First, we find the covariance of $Z_1$ and $Z_2$:
$Cov(Z_1, Z_2) = Corr(Z_1, Z_2) \sqrt{Var(Z_1) Var(Z_2)} = \frac{1}{3} \sqrt{1 \cdot 4} = \frac{1}{3} \cdot 2 = \frac{2}{3}$
Next, we find the covariance of $X$ and $Y$:
$Cov(X, Y) = Cov(Z_1 - Z_2 + 2, 4Z_1 + Z_2 - 2)$
$Cov(X, Y) = Cov(Z_1, 4Z_1) + Cov(Z_1, Z_2) + Cov(-Z_2, 4Z_1) + Cov(-Z_2, Z_2)$
$Cov(X, Y) = 4Cov(Z_1, Z_1) + Cov(Z_1, Z_2) - 4Cov(Z_2, Z_1) - Cov(Z_2, Z_2)$
$Cov(X, Y) = 4Var(Z_1) + Cov(Z_1, Z_2) - 4Cov(Z_1, Z_2) - Var(Z_2)$
$Cov(X, Y) = 4Var(Z_1) - 3Cov(Z_1, Z_2) - Var(Z_2)$
$Cov(X, Y) = 4(1) - 3(\frac{2}{3}) - 4 = 4 - 2 - 4 = -2$
Now, we find the variances of $X$ and $Y$:
$Var(X) = Var(Z_1 - Z_2 + 2) = Var(Z_1) + Var(-Z_2) + 2Cov(Z_1, -Z_2)$
$Var(X) = Var(Z_1) + Var(Z_2) - 2Cov(Z_1, Z_2) = 1 + 4 - 2(\frac{2}{3}) = 5 - \frac{4}{3} = \frac{15-4}{3} = \frac{11}{3}$
$Var(Y) = Var(4Z_1 + Z_2 - 2) = Var(4Z_1) + Var(Z_2) + 2Cov(4Z_1, Z_2)$
$Var(Y) = 16Var(Z_1) + Var(Z_2) + 8Cov(Z_1, Z_2) = 16(1) + 4 + 8(\frac{2}{3}) = 20 + \frac{16}{3} = \frac{60+16}{3} = \frac{76}{3}$
The correlation coefficient of $X$ and $Y$ is:
$Corr(X, Y) = \frac{Cov(X, Y)}{\sqrt{Var(X) Var(Y)}} = \frac{-2}{\sqrt{\frac{11}{3} \cdot \frac{76}{3}}} = \frac{-2}{\sqrt{\frac{836}{9}}} = \frac{-2}{\frac{\sqrt{836}}{3}} = \frac{-6}{\sqrt{836}}$
$Corr(X, Y) = \frac{-6}{2\sqrt{209}} = \frac{-3}{\sqrt{209}} = \frac{-3\sqrt{209}}{209}$
$Cov(X,Y) = 4(1)-3(2/3)-4 = 4-2-4 = -2$
$Var(X) = 1 + 4 - 2(2/3) = 5 - 4/3 = 11/3$
$Var(Y) = 16(1) + 4 + 8(2/3) = 20 + 16/3 = 76/3$
$Corr(X,Y) = \frac{-2}{\sqrt{(11/3)(76/3)}} = \frac{-2}{\sqrt{836/9}} = -6/\sqrt{836} = -3/\sqrt{209}$
$Corr(X,Y) = -3\sqrt{209}/209 \approx -0.2078$
$Cov(X,Y) = -2$.
$Corr(X,Y) \approx -0.2078$
Final Answer: The final answer is $\boxed{-2, -3/\sqrt{209}}$
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