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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·N-Kn-kNn

02

Calculation

So we have that

P(X=k+1)P(X=k)=Kk+1·N-Kn-(k+1)NnKk·N-Kn-kNn=Kk+1·N-Kn-(k+1)Kk·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)!·(N-K)!(n-k-1)!(N-K-n+k+1)!K!k!(K-k)!·(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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Most popular questions from this chapter

Five distinct numbers are randomly distributed to players numbered1through 5. Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players1and 2 compare their numbers; the winner then compares her number with that of player 3, and so on. Let X denote the number of times player 1 is a winner. FindPX=i,i=0,1,2,3,4.

If the distribution function of Xis given by

F(b)=0    b<012    0≤b<135    1≤b<245    2≤b<3910    3≤b<3.51    b≥3.5

calculate the probability mass function of X.

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability .3,and his second will lead independently to a sale with probability .6. Any sale made is equally likely to be either for the deluxe model, which costs \(1000, or the standard model, which costs \)500. Determine the probability mass function of X, the total dollar value of all sales

A box contains 5red and 5blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win \(1.10; if they are different colors, then you win -\)1.00. (That is, you lose $1.00.) Calculate

(a) the expected value of the amount you win;

(b) the variance of the amount you win.

Each of 500 soldiers in an army company independently has a certain disease with probability 1/103. This disease will show up in a blood test, and to facilitate matters, blood samples from all 500 soldiers are pooled and tested.

(a) What is the (approximate) probability that the blood test will be positive (that is, at least one person has the disease)? Suppose now that the blood test yields a positive result.

(b) What is the probability, under this circumstance, that more than one person has the disease? Now, suppose one of the 500 people is Jones, who knows that he has the disease.

(c) What does Jones think is the probability that more than one person has the disease? Because the pooled test was positive, the authorities have decided to test each individual separately. The first i − 1 of these tests were negative, and the ith one—which was on Jones—was positive.

(d) Given the preceding scenario, what is the probability, as a function of i, that any of the remaining people have the disease?
See all solutions

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