math - 確率と相対頻度

Question

事象の確率を推定するために相対頻度を使用する場合、実験の数に基づいた推定値はどれくらい良いでしょうか? 標準偏差は適切な尺度ですか? 紙/リンク/オンラインブックが最適です。

http://en.wikipedia.org/wiki/Frequentist

Answer 1

You count the number of successes s in a sequence n of Yes / No experiments, right? As long as the single experiments are independent you are in the realm of the Binomial distribution (Wikipedia). Frequency of success f = s / n is an estimator of the success probability p and. The variance of your frequency estimate f is p * (1-p) / n for n draws.

As long as p is not too close to zero or 1, and as long as you do not have "too small" a number of observations n, the standard deviation will be a reasonable measure for the quality of your estimate f.

If n is large enough (rule of thumb n * p > 10), you can approximate by a normal distribution N(f, f * (1-f) / n), and standard deviation estimate is a good measure. See here for a more extensive discussion.

This said the approximation with the standard deviation will not cut any ice if this needs to have some academic rigour (e.g. is a homework).