-1
$\begingroup$

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

$\endgroup$
1
$\begingroup$

Guide:

There are only two possibility, either they are identical or they are not.

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.

$\endgroup$
  • $\begingroup$ we need expected no. of distinct units , not identical , $\endgroup$ – ANUJ NAIN Jun 13 at 6:36
  • 1
    $\begingroup$ If you can compute the probability, you can get the expected value right? $\endgroup$ – Siong Thye Goh Jun 13 at 6:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.