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Chapter 4: Q. 4.29 (page 172)

For a hypergeometric random variable, determine

P{X=k+1}/P{X=k}

Short Answer

Expert verified

(K-k)(n-k)(k+1)(N-K-n+k+1)

Step by step solution

01

Given information

For Hypergeometric random variable with parameters N,K,nwe have that

P(X=k)=Kk路N-Kn-kNn

02

Calculation

So we have that

P(X=k+1)P(X=k)=Kk+1路N-Kn-(k+1)NnKk路N-Kn-kNn=Kk+1路N-Kn-(k+1)Kk路N-Kn-k

03

Continue Calculation

When we write out these binomial coefficients, we get that the expression above is equal to

K!(k+1)!(K-k-1)!路(N-K)!(n-k-1)!(N-K-n+k+1)!K!k!(K-k)!路(N-K)!(n-k)!(N-K-n+k)!

04

Final answer

If we cancel out everything we can we left with.

(K-k)(n-k)(k+1)(N-K-n+k+1)

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