## 4. Slices of a Normal Cake ## This problem needs only the material of a few weeks ago, but it's here because the ideas and visualization will be helpful in the next exercise. A former 140 staff member told me how he used this method in his interview at a major quant firm and surprised the interviewer. (Yes, he got the job.) Simple, insightful solutions tend to beat tons of calculus even when the calculus is done correctly. Let $X$ and $Y$ be independent standard normal random variables. **a)** Find $P(X > 0, Y > 0)$. Yes, it's easy. But get a piece of paper and draw the event on the plane anyway. Imagine the joint density surface over the plane, and try to imagine the relevant volume under the joint density surface as a quadrant-shaped slice of a bell-shaped cake. **Then use the same approach for the next two parts.** **b)** Find $P(X > 0, Y > X)$. **c)** Find $P(X > 0, Y > \sqrt{3}X)$