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SkipperLin · 2020年07月12日

问一道题：NO.PZ2016062402000016

  1. FRM I
  2. Quantitative

问题如下：

When can you use the normal distribution to approximate the Poisson distribution, assuming you have n independent trials, each with a probability of success of p?

选项：

A.

When the mean of the Poisson distribution is very small

B.

When the variance of the Poisson distribution is very small

C.

When the number of observations is very large and the success rate is close to 1

D.

When the number of observations is very large and the success rate is close to 0

解释：

The normal approximation to the Poisson improves when the success rate, χ, is very high. Because this is also the mean and variance, answers A and B. are wrong. In turn, the binomial density is well approximated by the Poisson density when np = χ is large.

请问这个success rate为什么要接近1呢 不是np越大就可以越好的approximate么

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品职答疑小助手雍 · 2020年07月12日

同学你好，它这里对success rate的定义是λ，而不是p。如果是p，那泊松分布下p应该是如课堂上所说，p趋近于0的。

这道题是2004年前的真题了。我们最近查了下原版书和notes，上面并没有这道题的结论。在泊松分布中，λ的定义是单位时间内事件的发生率（发生个数）。我们推测，之所以在04年时这道题选λ最大为1，是当时FRM参考书的定义问题。但我们现在都不这样说了，因为单位时间内事件发生的个数是可以大于1的。并且，我记得之前学概率论的时候，只是强调泊松分布中，λ=np的n足够大，就可以由中心极限定理趋近于正态分布。这道题有个印象就好，万一真的考到。但我们觉得是有点问题的。。

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