# Probability proportional to size

Consider the problem of drawing a sample of size $$2$$ from a finite population of size $$20$$ . The sampling is done with replacement using probability proportional to size scheme . The normed size measures $$p_1 , p_2 , \ldots, p_{20}$$ . are given by

$$p_i = \frac{1}{40} , i =1 ,\ldots,10$$

$$p_i = \frac{3}{40} , i =11 ,\ldots,20$$ , find the expected no. of distinct units drawn

attempt :

let $$X$$ denote the no. of distinct units drawn then $$X$$ denotes the value $$0 , 1 , 2$$ .

Now how to proceed to caculate probabilities , please provide some approach for this question or any alternative suggestion

For the probability of being identical, compute$$\sum_{i=1}^{20}P(X_2=i|X_1=i)P(X_1=i)=\sum_{i=1}^{20}P(X_2=i)P(X_1=i)$$
You might like to consider the sum for $$i$$ from $$1$$ to $$10$$ and the sum from $$11$$ to $$20$$ separately.