Expected Dart Score for a Bivariate Normal Throw with Ring Scoring

A dart hits the point $(X, Y)$, with $X$ and $Y$ independent normal random variables, both having mean 0 and variance $\sigma^2$. The chance that the dart falls within 1 unit of the origin equals $\tfrac{3}{4}$. A throw scores 16 points if the distance from the origin is ≤ 1, 4 points if the distance is > 1 but ≤ 2, and 0 points otherwise. What is the expected score for a single throw?

Hint — Think about the distance: Look at the distribution of the distance $R = \sqrt{X^2 + Y^2}$ from the origin. Determine whether you need the actual value of $\sigma$, or just a quantity that the given probability already fixes.

Constraints and Clarifications

• $\sigma > 0$ is not provided directly; it is determined by the condition $P(R \le 1) = 3/4$.
• The scoring regions are a disk of radius 1 and an annulus from radius 1 to 2. A point landing exactly on a boundary gets the higher score, but boundaries have probability zero.
• Provide the exact expected score.

What a Strong Answer Covers

• Deriving the radial distance distribution for independent, zero-mean normal coordinates with equal variance.
• Using the given probability to find the parameter that controls the scoring regions, without numerical approximation.
• Computing the ring's probability as the difference of two cumulative probabilities.
• Assembling the expected value and verifying that the region probabilities sum to 1.

Follow-up Questions

1. If the variance of each coordinate is doubled, what is the expected score?
2. Why does the calculation become more difficult if $X$ and $Y$ have different variances?
3. What is the variance of the score? Overview: An expected-value problem involving a dart whose landing point has independent zero-mean normal coordinates, calibrated such that it lands within 1 unit of the origin with probability 3/4. It asks for the expected score with 16 points inside radius 1 and 4 points up to radius 2, testing the radial distribution of a bivariate normal.