If coefficient of correlation between two variables X and Y is `0.64, cov(X, Y)= 16" and "Var(X)= 9`, then the standard deviation of Y series is

Options

A. `1.2`
B. `8.33`
C. `24`
D. `25`

Answer

To find the standard deviation of the Y series given the coefficient of correlation, covariance, and variance of X, we can follow these steps:

### Step 1: Identify the given values
- Coefficient of correlation (r) = 0.64
- Covariance (cov(X, Y)) = 16
- Variance of X (Var(X)) = 9

### Step 2: Calculate the standard deviation of X
The standard deviation (SD) of X can be calculated using the formula:
\[ \text{SD}(X) = \sqrt{\text{Var}(X)} \]
Substituting the given variance:
\[ \text{SD}(X) = \sqrt{9} = 3 \]

### Step 3: Use the formula for the coefficient of correlation
The formula for the coefficient of correlation is:
\[ r = \frac{\text{cov}(X, Y)}{\text{SD}(X) \times \text{SD}(Y)} \]
We can rearrange this formula to solve for the standard deviation of Y (SD(Y)):
\[ \text{SD}(Y) = \frac{\text{cov}(X, Y)}{r \times \text{SD}(X)} \]

### Step 4: Substitute the known values into the formula
Now, substituting the values we have:
\[ \text{SD}(Y) = \frac{16}{0.64 \times 3} \]

### Step 5: Calculate SD(Y)
First, calculate the denominator:
\[ 0.64 \times 3 = 1.92 \]
Now substitute this back into the equation:
\[ \text{SD}(Y) = \frac{16}{1.92} \]
Calculating this gives:
\[ \text{SD}(Y) \approx 8.33 \]

### Final Result
Thus, the standard deviation of the Y series is approximately \( \frac{25}{3} \) or \( 8.33 \).

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