4. (10%) Suppose that 500 points are selected independently and at random from the unit square x, y): 0 x < 1, 0 y < 1\. Let $W$ equal the number of points that fall into \(x, y): x^2 + y^2 < 1\. (a) (5%) Write down the distribution of \( W and the corresponding probability mass function.

9. (10%) In a college health fitness program, let $X$ denote the weight in kilograms of a male freshman at the beginning of the program and $Y$ denote the weight change during a semester. Assume $X$ and $Y$ have a bivariate normal distribution with $\mu_X = 75$ and $\mu_Y = 5$, $\sigma_X^2 = 64$, $\sigma_Y^2 = 2.56$ and $\rho = -0.6$. The joint probability density function of a bivariate normal distribution is defined as $$f(x, y) = \frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}} \exp\left(-\frac{q(x, y)}{2}\right)$$ where $q(x, y) = \frac{1}{1-\rho^2} \left[ \left(\frac{x-\mu_X}{\sigma_X}\right)^2 - 2\rho \left(\frac{x-\mu_X}{\sigma_X}\right) \left(\frac{y-\mu_Y}{\sigma_Y}\right) + \left(\frac{x-\mu_Y}{\sigma_Y}\right)^2 \right]$. (a) Derive the marginal pdf of $ Y $ and find $ P[2.5 \leq Y \leq 6.5] $. [A detailed derivation is required.]

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