Financial Risk Manager Part 1

The correct answer is B) 0.657.

Explanation:

This problem involves calculating a probability from a joint probability density function. The solution requires:

1. Finding the marginal probability density function of Y from the joint PDF f(x,y) = 6(1 - (x + y)) for 0 < x < 1-y, 0 < y < 1.

2. The marginal PDF of Y is calculated as: $f_Y(y) = \int_0^{1-y} 6(1 - (x + y)) \, dx$ $= \left[6 \left(x - \frac{x^2}{2} - xy\right)\right]_0^{1-y}$ $= 6\left[1 - y - \frac{(1-y)^2}{2} - y(1-y)\right]$ = 3 - 6y + 3y^2$`$3`. **The probability P(Y < 0.3) is then:** P(Y < 0.3) = \int_0^{0.3} (3 - 6y + 3y^2) , dy$$= \left[3y - 3y^2 + y^3\right]_0^{0.3}$$= 3(0.3) - 3(0.3)^2 + (0.3)^3$$= 0.9 - 0.27 + 0.027$$= 0.657$$

This is a typical problem in probability theory involving joint distributions and marginal distributions, which falls under the Quantitative Analysis section of the FRM curriculum.