SOLUTION: Given a random variable X, with standard deviation σX, and a random variable Y = a + bX, show that if b < 0, the correlation coefficient ρXY = −1, and if b

SOLUTION: Given a random variable X, with standard deviation σX, and a random variable Y = a + bX, show that if b < 0, the correlation coefficient ρXY = −1, and if b > 0, ρXY = 1

Question 1204179: Given a random variable X, with standard deviation σX, and a random variable Y = a + bX, show
that if b < 0, the correlation coefficient ρXY = −1, and
if b > 0, ρXY = 1
Answer by ElectricPavlov(122) (Show Source): You can put this solution on YOUR website!
**1. Find the Standard Deviation of Y**
* **Y = a + bX**
* **σY = |b| * σX**
* The absolute value of 'b' is used because standard deviation must be non-negative.
**2. Find the Covariance of X and Y**
* **Cov(X, Y) = Cov(X, a + bX)**
* **Cov(X, Y) = Cov(X, a) + Cov(X, bX)**
* **Cov(X, Y) = 0 + b * Var(X)**
* Cov(X, a) = 0 because the covariance between a random variable and a constant is zero.
* Cov(X, bX) = b * Var(X)
* **Cov(X, Y) = b * σX²**
**3. Calculate the Correlation Coefficient (ρXY)**
* **ρXY = Cov(X, Y) / (σX * σY)**
* **ρXY = (b * σX²) / (σX * |b| * σX)**
* **ρXY = b / |b|**
**4. Determine the Sign of ρXY**
* **If b > 0:**
* ρXY = b / b = 1
* This indicates a perfect positive linear relationship between X and Y.
* **If b < 0:**
* ρXY = b / (-b) = -1
* This indicates a perfect negative linear relationship between X and Y.
**Therefore:**
* If b < 0, ρXY = -1
* If b > 0, ρXY = 1
This demonstrates that the correlation coefficient between X and Y is perfectly positively or negatively correlated depending on the sign of the slope (b) in the linear relationship Y = a + bX.

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