Expected Dart Score for a Bivariate Normal Throw with Ring Scoring

A dart lands at the point $(X, Y)$, where $X$ and $Y$ are independent normal random variables, each with mean 0 and variance $\sigma^2$. The probability that the dart lands within 1 unit of the origin is $\tfrac{3}{4}$. A throw scores 16 points if its distance from the origin is at most 1, 4 points if its distance is greater than 1 but at most 2, and 0 points otherwise. What is the expected score of a single throw? ```hint Think in terms of distance Consider the distribution of the distance $R = \sqrt{X^2 + Y^2}$ from the origin. Ask whether you need the value of $\sigma$ itself, or only a quantity that the given probability already determines. ``` ### Constraints and Clarifications - $\sigma > 0$ is not given directly; it is fixed by the condition $P(R \le 1) = 3/4$. - The scoring regions are a disk of radius 1 and the ring between radii 1 and 2. A point exactly on a boundary circle receives the higher score, although boundaries have probability zero. - Give the exact expected score. ### What a Strong Answer Covers - Deriving the distribution of the radial distance for independent, zero-mean normal coordinates with equal variance. - Using the calibration probability to determine the parameter the scoring regions depend on, without numerical approximation. - Computing the probability of the ring as a difference of cumulative probabilities. - Assembling the expected value and checking that the region probabilities sum to 1. ### Follow-up Questions 1. What is the expected score if the variance of each coordinate is doubled? 2. Why does the calculation become harder if $X$ and $Y$ have different variances? 3. What is the variance of the score?