Estimate the probability three uniforms sum below one

Question

Estimate P(X + Y + Z < 1) where X, Y, Z are independent U(0,1). Geometrically this is the volume of the corner tetrahedron in the unit cube, whose true value is 	frac{1}{6} \approx 0.1667. Write the estimator, and size the sample for roughly 3 correct decimals.

Accepted Answer

Each trial is a Bernoulli hit: draw three uniforms and test the inequality. The fraction of hits estimates the probability: Convergence. The hit count is Binomial(n, p) with p = 1/6, so the standard error is \sigma = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{(1/6)(5/6)}{n}} \approx \frac{0.373}{\sqrt n}. For error near 10^{-3} you need n \approx (0.373/0.001)^2 \approx 1.4 	imes 10^{5} samples. Put a confidence interval on it with \hat p \pm 1.96\,\hat\sigma. Takeaway. Estimating a probability is a Bernoulli Monte Carlo: count hits, divide by n, and read the error straight off the binomial stand…

Estimate the probability three uniforms sum below one

Your answer

More Coding questions