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Classic setup -- given some coin with $P(H) = 1-q$ where $q$ is some random variable (RV) with $$f_Q(q)= 2q, 0 \le q \le 1$$ and $0$, otherwise. Assume conditioned on $Q$, each coin flip is independent.

  1. Calculate P(H) given single toss, and
  2. given some RV, $Y_i$ where $Y_i=1$ when coin toss is H, and $0$ otherwise. There are 30 days and within each day a coin is tossed, find $Var(Z)$ where $Z=4(Y_1 + ...+ Y_{30})$.

I have no problems with 1) having $p_{H|Q}(h|q)$ and $f_Q(q)$ I find joint $f_{H,Q}(h,q)$ and integrate over $q$., so $p(H) = 1/3$. I'm struggling with 2) -- I perfectly understand that I can find answer via law of total variance, namely $Var(Z) = Var(E(Z|Q)) +E(Var(Z|Q))$, but why I can't straight substitute value obtained in 1) and look at $Y_i \sim Ber(\frac{1}{3})$? What is the difference between these approaches? I though that via 1) I transform problem into sum of 30 iid Bernoulli rv's. But it seems not to be the case...

Thanks in advance.

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$Y_i\sim \text{Ber}(1/3)$ is correct, but $Y_i$ are not independent unless $Q$ is given. So, you can't distribute variance calculation over the sum. This dependence is broken inside the total variance formulas when $Q$ is assumed to be given.

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  • $\begingroup$ Thank you. But how $Y_i$ are dependent? Before coin toss I obtain it's bias and it stays the same for all 30 tosses. Also Q is given and is 1/3 (or 2/3). $\endgroup$ – Sharov Oct 30 '20 at 0:09
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    $\begingroup$ No, when $Q$ is given $Y_i | Q$ is Ber(1-Q), but when it's not given, $Y_i$ is Bern with $1/3$. Q is not $2/3$ here. Hence, $Y_i$ are dependent since they have underlying Q common. $\endgroup$ – gunes Oct 30 '20 at 0:18
  • $\begingroup$ Am I right that Q here is something like hidden\unobservable\latent variable? And so all $Y_i$ is correlated in this case? Once we explicitly depend on it they are no more correlated,correct? $\endgroup$ – Sharov Oct 30 '20 at 8:39
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    $\begingroup$ Yes, and more precisely, once it's explicitly stated as given, $Y_i$ are not correlated any more. $\endgroup$ – gunes Oct 30 '20 at 8:55

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