Let [tex]X[/tex] be the set [tex]X=\left\{x_1,x_2,...,x_n\right\}[/tex], where each [tex]x_i[/tex] is a Bernoulli random variable and [tex]P[/tex] be the set [tex]P=\left\{p_1,p_2,...,p_n\right\}[/tex], where [tex]p_i[/tex] is the probability that [tex]x_i=1[/tex]. Now, suppose there are two other sets, [tex]A=\left\{a_1,a_2,...,a_n\right\}[/tex] and [tex]B=\left\{b_1,b_2,...,b_n\right\}[/tex], where [tex]a_i,b_i[/tex] are two different estimates for [tex]p_i[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

In other words, p is the "true" (unknown) probability that x will occur, and a and b are both attempts to estimate that probability.

I am trying to design an experiment that will determine which set, A or B, is "closer" to P.

If these Bernoulli random variables represented, say, different-sided dice, then an experiment would be pretty straightforward: conduct a Bernoulli process; i.e., repeatedly perform a Bernoulli trial.

However, in my case, each [tex]x_i[/tex] represents a real-world event, and can be simulated only once. A Bernoulli process is impossible, I am limited to a single Bernoulli trial. If [tex]X=\left\{x_1,x_2,x_3\right\}[/tex], then I don't think there would be a practical way to solve this problem at any level of significance. But [tex]n[/tex] here is actually quite large, so I feel like there should be some test which would allow me to show [tex]A>B[/tex] or [tex]A<B[/tex] or [tex]A\neq B[/tex] or [tex]A=B[/tex].

Any thoughts?

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# Bernoulli Trials

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