If `cov(X, Y)= 16.5, Var(X)= 8.25" and "Var(Y)= 33`, then Karl Pearson's coefficient of correlation between X and Y is

Options

A. `(1)/(2)`
B. `(2)/(3)`
C. `(3)/(4)`
D. `1`

Answer

To find Karl Pearson's coefficient of correlation \( r \) between two variables \( X \) and \( Y \), we can use the formula:

\[ r = \frac{cov(X, Y)}{\sqrt{Var(X) \cdot Var(Y)}} \]

### Step-by-step Solution:

1. **Identify the given values**:
- Covariance \( cov(X, Y) = 16.5 \)
- Variance \( Var(X) = 8.25 \)
- Variance \( Var(Y) = 33 \)

2. **Substitute the values into the formula**:
\[ r = \frac{16.5}{\sqrt{8.25 \cdot 33}} \]

3. **Calculate the product of the variances**:
- First, multiply \( Var(X) \) and \( Var(Y) \):
\[ 8.25 \cdot 33 = 272.25 \]

4. **Take the square root of the product**:
\[ \sqrt{272.25} = 16.5 \]

5. **Substitute back into the formula**:
\[ r = \frac{16.5}{16.5} \]

6. **Simplify the expression**:
\[ r = 1 \]

### Final Answer:
The Karl Pearson's coefficient of correlation \( r \) between \( X \) and \( Y \) is \( 1 \).