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I'm a little confused how the CLT can apply to aggregations of the exponential distribution. It's my understanding that the CLT says, in plain English, "sample means from virtually any distribution will tend towards normal." I've seen this demonstrated in the case of rolling a fair, 6-sided die. If single rolls are sampled then the distribution will remain uniform. However, if 2+ rolls were executed per trial, then the resulting distribution will be approximately normal, peaking at 3.5 when n is sufficiently high.

However, I'm trying to replicate this process with the exponential distribution and not finding the result I was expecting.

import numpy as np
import matplotlib.pyplot as plt

samples = []
sample_size = 2
itrs = 10_000
while len(samples) < itrs:
  samples.append(sum(np.random.exponential(scale=1,size=sample_size))/sample_size)

plt.hist(dist,bins=12)

exponential histogram

So either my understanding of the CLT is far off or the numpy's exponential sampler doesn't behave the way that I had expected. Could someone clarify where I'm going wrong?

jbuddy_13
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    Your sample size of $2$ is scarcely large! Keep increasing it and study the output. – whuber Aug 19 '21 at 18:32
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    The CLT is actually about the limiting distribution of standardized averages or sums *in the limit* as $n\to\infty$, not at any sample size you like. Now you can often see averages or sums start to look pretty close to normal at modest sample sizes, but for exponentials $n=2$ is just not close enough to infinity. If you want averages to look quite close to normal at sample sizes of 2 you need to start fairly close to normal. For some distributions (for which the CLT still works), sample sizes in the billions are not sufficient. – Glen_b Aug 19 '21 at 18:33
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    There's quite a few answers on site that address aspects of this. – Glen_b Aug 19 '21 at 18:37
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    https://stats.stackexchange.com/questions/339759/how-can-the-central-limit-theorem-hold-for-distributions-which-have-limits-on-th and https://stats.stackexchange.com/questions/422870/central-limit-theorem-on-distributions-with-support-other-than-mathbbr – Dave Aug 19 '21 at 18:41

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