Below I have a copy of a Chegg solution. I do not understand how the textbook so

Question

Below I have a copy of a Chegg solution. I do not understand how the textbook solution got E[Y]=(N/n)Y

Is there a less complicated way to do this? I never learned what this solution is saying.

An urn contains e black balls and N- e white balls. A sample of n balls is to be selected without The first population moment (row moment) of Yis given by replacement. Let Y denote the number of black balls in the sample. Show that (N n)Y is the nts estimator of Given that an urn contains 0black balls and N-0white balls and a sample of nballs is y selected without replacement. Ydenotes the number of black balls in the sample. Then Y follows hypergeometric distribution with parameters n, N,0 and its probability mass o y!(n-y)! (N)n function is given by, 600-1 n (n-1) y N(N-1 n-1 (0) n -0 (n -1) 0.1.22 p( In,N,0) y (Where n n-1 y y-1] elsewhere (N-1) 0-1)( n-0 n60 n Y y Since Y' Hypergeometric (n',N 1,0-1) nb so the sum of its probability mass function over full range is unity

Explanation / Answer

tne solution doen is very nice ,you just have to know the properties of binomial coefficients , and property of pmf (probability mass function) that summation of all probability = 1.

there are other website which can be useful

http://www.math.uah.edu/stat/urn/Hypergeometric.html

see property no 8)

http://faculty.madisoncollege.edu/alehnen/EngineeringStats/hypergdistribution.pdf

https://math.stackexchange.com/questions/1380460/derivation-of-mean-and-variance-of-hypergeometric-distribution

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