Answer: 0.657

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.

Author: Nikitesh Somanthe