• Correlation vs. Independence in Standard Normal Variables:
• Given $X \sim N(0, 1)$ and $Y = X^2$:
• The Pearson correlation coefficient is given by:

$$\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} = \frac{E[XY] - E[X]E[Y]}{\sigma_X \sigma_Y}$$

• For a standard normal distribution:
• $E[X] = 0$
• $E[XY] = E[X \cdot X^2] = E[X^3]$
• Since the standard normal distribution is symmetric about 0, all its odd central moments are zero:

$$E[X^3] = 0$$

• Therefore, the covariance is:

$$\text{Cov}(X, Y) = E[X^3] - E[X]E[X^2] = 0 - 0 = 0$$

• Consequently, $\rho(X, Y) = 0$.
• Key Insight: A correlation coefficient of 0 indicates absence of linear association, even though $X$ and $Y$ have a perfect non-linear functional dependence ($Y = X^2$) (Option 2).

Correct option: (2) 0