Win rates at first-passage times for biased simple random walks

TL;DR

This paper analyzes the win rate of a biased simple random walk at the first-passage time, deriving explicit formulas and proposing improved unbiased estimators for π based on first-passage sampling.

Contribution

It provides explicit formulas for the expectation and variance of the win rate and introduces novel unbiased estimators of π using biased coin-flip sampling.

Findings

Explicit formulas for expectation and variance of win rate.

Monotonicity properties in bias and threshold.

Unbiased estimators of π with improved accuracy and efficiency.

Abstract

We study the win rate $R_{N_d}/N_d$ of a biased simple random walk $S_n$ on $\mathbb{Z}$ at the first-passage time $N_d=\inf\{n\ge 0:S_n=d\}$, with $p=P[X_1=+1]\in[1/2,1)$. Using generating-function techniques and integral representations, we derive explicit formulas for the expectation and variance of $R_{N_d}/N_d$ along with monotonicity properties in the threshold $d$ and the bias $p$. We also provide closed-form expressions and use them to design unbiased coin-flipping estimators of $\pi$ based on first-passage sampling; the resulting schemes illustrate how biasing the coin can dramatically improve both approximation accuracy and computational cost.