Simple random sampling.
Hey guys! :D
I need some help and hopefully you can help me! :) And my apologies for not using latex - I am a total noob with it! My problem is:
A simple random sample of size n = 3 is selected with replacement from a population of size N = 10 whose mean and variance are Ybar = 26 and S^2 = 39.
a) Find the probabilities that the sample contains 2 and 3 distinct units respectively.
And I have been given the hint that the probability that the sample contains 1 distinct unit is p_1=1/(N^2) =0.01
Okay, for starts, regarding the hint, how was the p_1 derived? and how do we approach such a question? I don't want answers, but I really need guidance from someone out there as to where to start on this because I am really struggling at the moment! :( Thanks for your help in advance! :)
Probability of getting 1 distinct unit in 3 selections means in all the 3 selections, only 1 particular element is drawn.
Now, for the first selection, whatever element we choose it will be distinct. So the probability is N/N = 1.
For 2nd selection, that particular element (which was drawn in the 1st selection) has to be drawn out of N elements. So the probability is 1/N.
Similarly for 3rd selection, the probability will also be 1/N.
So the final probability becomes the product of the above three --- which is 1/N^2.
try to find your answer now.
Thank you for your help! I wasn't expecting a reply :)
So, if I may ask, whether if my answers and reasonings for this question are correct:
p_2=N/N x (N-1)/N x 2(N-1)/N = 3(N-1)/N
The bit I was not sure of was the '2(N-1)/N' part - does that make sense? I can make sense of it in my head but it is hard to express and I am not sure if I am right at all. Next,
p_3= N/N x (N-1)/N x (N-2)/N = (N-1)(N-2)/N
which I am pretty sure is right.
This question also has a part (b): d-bar is the arithmetic mean of the distinct observations in the sample. Show that d-bar is unbiased.
I have been given the hint that we should first find the expected value of d given that the sample contains 1 distinct unit.
Regarding the hint, is that simply:
expected value of d given that the sample contains 1 distinct unit = N times p_1