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Expected Dart Score for a Bivariate Normal Throw with Ring Scoring

DRWQuantitative Researcher · Onsite
HardQuantitative ResearcherOnsiteStatistics & Math

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Calculate the expected score of a dart throw modeled by a zero-mean bivariate normal distribution with circular scoring rings, calibrated using a known hit probability inside the inner ring.

Problem Statement

A quantitative trader plays a dart game where the landing position of a dart, (X, Y), follows a bivariate normal distribution centered at the origin (0, 0). Specifically, X and Y are independent and identically distributed normal random variables:

X ~ N(0, sigma^2) Y ~ N(0, sigma^2)

Let R = sqrt(X^2 + Y^2) denote the Euclidean distance from the landing point to the center of the board.

The thrower's accuracy is calibrated such that the probability of landing within radius 1 of the origin is exactly 3/4:

P(R <= 1) = 3/4

The dartboard has two concentric scoring rings centered at the origin:

  • Bullseye / Inner Ring (R 2): awards 0 points.

Tasks

  1. Express the cumulative distribution function P(R <= r) in terms of sigma.
  2. Determine the exact value of P(R <= 2) without needing to numerically approximate sigma.
  3. Compute the expected score of a single throw as an exact fraction in simplest form.

Hints

Hint 1

Recall that the sum of squares of two independent standard normal variables follows a chi-squared distribution with 2 degrees of freedom, which is identical to an Exponential distribution with scale parameter 2.

Hint 2

Express P(R <= r) as 1 - exp(-r^2 / (2 * sigma^2)), and use the calibration condition P(R <= 1) = 3/4 to directly find the value of exp(-1 / (2 * sigma^2)).

Hint 3

Notice that exp(-2^2 / (2 * sigma^2)) is simply (exp(-1 / (2 * sigma^2)))^4, allowing you to compute P(R <= 2) and P(1 < R <= 2) algebraically.

Follow-up questions

  • How does the radial distribution change if X and Y have unequal variances sigma_x^2 != sigma_y^2, and how would you calculate the hit probability in that scenario?
  • If the thrower has an aim bias (mu_x, mu_y) != (0, 0), what is the distribution of R^2, and does offsetting the aim ever increase expected score given these ring values?
  • Suppose the scoring rule is continuous, granting S(R) = max(0, 16 - 4 * R^2) points. What would the expected score be under the same calibration?