4-9. Negative Hypergeometric Distribution (*)
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6κ°μ ν° κ³΅, 4κ°μ κ²μ κ³΅μ΄ λ μμμμ 곡μ νλμ© κΊΌλΌ λ, ν° κ³΅μ κΊΌλ΄λ κ²μ 'μ±κ³΅'μ΄λΌκ³ νλ€λ©΄, ν° κ³΅μ μ²μ κΊΌλ΄κΈ° μ κ²μ 곡μ κΊΌλΈ νμλ₯Ό νλ₯ λ³μ λΌ ν λ, μ΄ νλ₯ λ³μλ₯Ό 'Negative Hypergeometric random variable' μ΄λΌκ³ νλ€.
52μ₯μ νλ μ μΉ΄λλ₯Ό κ³ λ£¨ μκ³ , ν μ₯μ© λλ μ€λ€κ³ μκ°ν΄λ³΄. μ¬κΈ°μ 'μμ΄μ€' μΉ΄λκ° λμ€λ©΄ μ±κ³΅μ μ΄λ€. λ§μ½ 7λ²μ§Έμμ λλ μ€ μΉ΄λκ° 'μμ΄μ€'μΉ΄λ μλ€λ©΄, μ΄ μ΄μ μ λλ 쀬λ 6κ°μ μΉ΄λλ€μ 'Negative Hypergeometric Distribution'μ λ°λ₯΄λ κ²μ΄λ€.
One can view the Binomial distribution and Hypergeometric distribution as both considering a random variable that counts the number of successes in trials.
In the Binomial case, there are Bernoulli trials with constant probability of success from trial to trial, with independent trials.
In the Hypergeometric case, the sampling is without replacement so that probabilities change from selection to selection and trials are dependent. In the Bernoulli trials case, the Negative Binomial distribution is the distribution counting the number of trials required until a specified number (say ) of successes have been observed. In the sampling without replacement case, a similar situation is to consider the number of selections required until a success is obtained.
EXAMPLE 29. Suppose an urn contains 4 red and 10 blue balls and that balls are drawn one after another from this urn until a red ball is obtained. What is the probability that exactly six balls are drawn?
[ Solution ]
.
This example is like "waiting for the first success in sampling without replacement" with success being obtaining a red ball.
Recall that in sampling with replacement the distribution analogous to this was the Geometric distribution, a special case of the Negative Binomial distribution.
EXAMPLE 30. A statistics department has purchased 24 calculators of which 4 are defective. Calculators are selected one-after-another without replacement and tested. What is the probability that the second calculator found to be defective is the eighth calculator selected?
[ Solution ]
In the above two examples, the final selection must be a success. The selections before this one simply involve a Hypergeometric situation involving one fewer selection than the total number and one fewer success than the number for which the procedure is waiting.
Negative Hypergeometric Distribution
Comments:
Again, it might be harder to try to remember the formula for the Negative Hypergeometric probability function than to simply solve the problem based on general knowledge of probability.
EXAMPLE 31. A land developer has plans for having 86 acreages in its development south of the city. During the development of the acreages, water testing has suggested that 12 of the sites have water problems such that the wells on these sites do not have water that meets local drinking standards. If a potential purchaser decides to visit several of the acreage sites chosen at random from the 86, what is the probability that the third site that the purchaser visits that has such water problems is the eighth site visited? What is the expected number of sites that this purchaser would visit so as to have found three with such water problems?
[ Solution ]
μ§κΈκΉμ§ μμ λ΄€λ 1. μ΄νλΆν¬, 2. μ΄κΈ°νλΆν¬, 3. μμ΄νλΆν¬, 4. Negative Hypergeometric Distribution λ₯Ό ν λ² μκ°ν΄ 보μ.
μ΄νλΆν¬λ μννμ(number of trials)κ° μ ν΄μ Έ μμλ€. μλ₯Ό λ€μ΄, μνλ¬Έμ 25λ¬Έμ λ₯Ό μ°μ΄μ λ€ λ§ν νλ₯ μ²λΌ '25'λΌλ μννμκ° κ³ μ λμ΄ μλ€.
μ΄κΈ°νλΆν¬λ λ§μ°¬κ°μ§ μ΄λ€. 1,000κ°μ μ ν μ€ 20κ°λ₯Ό λ½λ κ²μ²λΌ '20'μ΄λΌλ μννμλ₯Ό μ ν΄λκ³ λ¬Έμ λ₯Ό νΌλ€.
νμ§λ§, μ΄νλΆν¬μ μ΄κΈ°νλΆν¬μ μ°¨μ΄μ μ '볡μμΆμΆ(with replacement)'μ΄λ 'λΉλ³΅μμΆμΆ(without replacement)'μ΄λμ μ°¨μ΄μ΄λ€. μ΄νλΆν¬λ 볡μμΆμΆ, μ΄κΈ°νλΆν¬λ λΉλ³΅μμΆμΆμ μ μ λ‘ νλ€.
μμ΄νλΆν¬λ μννμλ₯Ό μ ν΄ λμ κ²μ΄ μλλΌ 'μ±κ³΅'μ νμ(number of successes)λ₯Ό μ ν΄ λκ³ λ¬Έμ λ₯Ό νΌλ€. μ¦ μλμ리μ¦μμ 4λ² λ¨Όμ μΉλ¦¬νλ νμ΄ μ°μΉνλ κ²μ΄ μκ° λλ€.
Negative Hypergeometic Distribution λ μ±κ³΅νμλ₯Ό μ ν΄λμ΅λλ€. 첫 λ²μ§Έ μ±κ³΅μ΄ λμ€κΈ° μ κΉμ§ λͺ λ²μ μ€ν¨λ₯Ό κ±°λν΄μΌ νλμ§λ₯Ό μμ보λ κ²μ΄λ€.
μ΄ λ΄μ©μ νλ‘ μ 리νλ©΄ λ€μκ³Ό κ°λ€.
With Replacement
Without Replacement
Fixed Number of Trials
μ΄ν λΆν¬
(Binomial Distribution)
μ΄κΈ°ν λΆν¬
(Hypergeometric Distribution)
Fixed Number of Successes
μμ΄ν λΆν¬
(Negative Binomial Distribution)
Negative Hypergeometric
Distribution
.
Let be a random variable counting the number of selections required until the success is obtained when sampling without replacement from a set of objects of which have a certain attribute (i.e. success). Then is said to have a Negative Hypergeometric distribution with parameters and -- that is, -- and, for appropriate values , its probability function is
where, the parameters and are non-negative integers which satisfy the condition .
In the above expression, the quantity is just the Hypergeometric probability if (i.e. exactly successes in the first draws), and the second is the probability of another success on the draw based on what remains of the set of objects.
The value space of this random variable is .
The mean of this distribution is and its variance is .
Let be a random variable counting the number of sites visited up to and including the third one having these water problems. Then . The probability of having to visit 8 sites is . And .