Question

Let \(X\) be a \(U(0,1)\) random variable and \(Y = X^2\). If \(\rho\) is the correlation coefficient between \(X\) and \(Y\), then \(48\rho^2\) is equal to:

• A. 48
• B. 45
• C. 35
• D. 30

Hint:

For correlation between \( X \) and \( X^2 \), use known moments of the uniform distribution and simplify using definitions of covariance and variance.

Answer 1

The Correct Option is B

Step 1: Compute expectations.
For \( X \sim U(0,1) \): \[ E[X] = \frac{1}{2}, \quad E[X^2] = \frac{1}{3}, \quad E[X^3] = \frac{1}{4}, \quad E[X^4] = \frac{1}{5} \]
Step 2: Compute covariance.
\[ \text{Cov}(X,Y) = E[XY] - E[X]E[Y] = E[X^3] - E[X]E[X^2] = \frac{1}{4} - \frac{1}{2}\times\frac{1}{3} = \frac{1}{12} \]
Step 3: Compute variances.
\[ \text{Var}(X) = E[X^2] - (E[X])^2 = \frac{1}{3} - \frac{1}{4} = \frac{1}{12} \] \[ \text{Var}(Y) = E[X^4] - (E[X^2])^2 = \frac{1}{5} - \frac{1}{9} = \frac{4}{45} \]
Step 4: Compute correlation coefficient.
\[ \rho = \frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)\text{Var}(Y)}} = \frac{\frac{1}{12}}{\sqrt{\frac{1}{12}\cdot\frac{4}{45}}} = \frac{\sqrt{15}}{8} \] \[ 48\rho^2 = 48 \times \frac{15}{64} = 45 \] Final Answer: \[ \boxed{45} \]