Home » Basic probability » Limit Theorems » Limit for mixtures

Limit for mixtures

Consider the following mixture distribution.

  1. Draw \(X \sim \mbox{Be}(p=.3)\)
  2. If \(X=1\) then \(Y \sim \mbox{Geo}(p_1)\)
  3. 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 \(p\). )

Topics