Breaking down a summation

Main Question or Discussion Point

Can someone help me break this down?

[tex]\Sigma^{k}_{i=1}\frac{i \left(^{n}_{i}\right)\left(^{m}_{k-i}\right)}{\left(^{m+n}_{k}\right)}[/tex]
 

Answers and Replies

k*n/(m+n)
 
Thanks for your help, but I had the answer and was really looking for the process.
 
ssd
268
6
Look at the expectation of a Hypergeometric variable.
 
111
1
Can someone help me break this down?

[tex]\Sigma^{k}_{i=1}\frac{i \left(^{n}_{i}\right)\left(^{m}_{k-i}\right)}{\left(^{m+n}_{k}\right)}[/tex]

First translate from math to English: there are m red balls and n blue balls in a sack from which you randomly draw k balls. What is the expected number of blue balls drawn? Now translate back into math: try using indicator random variables [itex]X_{j}[/itex] which equal 1 if the j-th drawn ball is blue and 0 if it is red. Now define the random variable

[tex]
X = \sum_{j=1}^{k} X_{j}
[/tex]

and compute the expected value of that and hopefully you'll get the answer that Roberto gave.


addendum: doh! After all that I just realized you can factor the answer out of the sum. Then use the illustration of selecting balls to see what the resulting sum must be.
 
Last edited:

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