Consider the following mixture distribution.
- Draw \(X \sim \mbox{Be}(p=.3)\)
- If \(X=1\) then \(Y \sim \mbox{Geo}(p_1)\)
- If \(X= 0\) then \(Y \sim \mbox{Bin}(n,p_2)\)
Consider the sequence of random variables \(Y_1,…,Y_{200}\) drawn iid from the above random experiment.
Use the central limit theorem to state the distribution of \(S = \frac{1}{200} \sum_i^{200} Y_i\).
(Here \(\mbox{Be}(p)\) is the Bernoulli distribution with parameter \(p\) and \(\mbox{Geo}(p)\) is the geometric distribution with the parameter \(p\). )