[LaserPineapple]

> The sum or sample average of a simple mixture of Bernoulli distributions still converges to a Normal.

The procedure given in the example is not a sum of samples from a mixture of Bernoulli distributions. It is a mixture of sums of Bernoulli distributions.

[em500]

The writing is not very clear, but I understand it to be a mixture of Bernoulli RVs:

> Let's consider the following scenario: say we have three coins with different biases (their probability of coming up heads): 0.4, 0.5 and 0.6. We pick one of the three coins at random, toss it 300 times and count the number of heads. What is the distribution obtained?

This is approximating the sum of 300 Bernoulli RVs with a Normal, which is perfectly valid.

> As we have seen, if we fix the coin we're tossing, the number of heads can effectively be approximated by a distribution N(300p,300p(1−p)) (where p is the coin's bias). This time, however, each time we take a sample we might be tossing any of the three different coins.

I understand the procedure here to be (1) choosing one of the 3 coins at random and tossing it, (2) repeating step (1) 300 times and summing the resulting number of heads. In this case the CLT does apply: the distribution of the sum-of-number-of-heads is approximately Normal, not the plotted tri-modal density.

[LaserPineapple]

> We pick one of the three coins at random, toss it 300 times and count the number of heads.

This is the procedure. One of the coins is chosen, and that same coin is flipped 300 times. Conditional on which coin was chosen, the number of heads is Binomial, so the unconditional distribution is a mixture of three Binomials.

[em500]

But a few lines later, it states

> This time, however, each time we take a sample we might be tossing any of the three different coins.

which is a different procedure