الرياضيات

عدد المواضيع في هذا القسم 7985

تاريخ الرياضيات

الرياضيات المتقطعة

الجبر

الضبابية

نظرية المجموعات

نظرية الزمر

نظرية الحلقات والحقول

نظرية الاعداد

نظرية الفئات

حساب المتجهات

المتتاليات-المتسلسلات

المصفوفات و نظريتها

المثلثات

الهندسة

التفاضل و التكامل

المعادلات التفاضلية و التكاملية

التحليل

التبلوجيا

نظرية الالعاب

الاحتمالات و الاحصاء

نظرية التحكم

بحوث العمليات

نظرية الكم

الشفرات

مواضيع عامة في الرياضيات التطبيقية

نظريات ومبرهنات

علماء الرياضيات

الرياضيات في العلوم الاخرى

بحوث و اطاريح جامعية

هل تعلم

طرائق التدريس

الرياضيات العامة

نظرية البيان

الرياضيات : الاحتمالات و الاحصاء :

Hypergeometric Distribution

02:52 AM

18 / 4 / 2021

Author : Beyer, W. H.

Book or Source : CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press

page : ...

Hypergeometric Distribution

Let there be ways for a "good" selection and ways for a "bad" selection out of a total of n+m possibilities. Take samples and let x_i equal 1 if selection is successful and 0 if it is not. Let be the total number of successful selections,

(1)

The probability of successful selections is then

			(2)
			(3)
			(4)

The hypergeometric distribution is implemented in the Wolfram Language as HypergeometricDistribution[N, n, m+n].

The problem of finding the probability of such a picking problem is sometimes called the "urn problem," since it asks for the probability that out of balls drawn are "good" from an urn that contains "good" balls and "bad" balls. It therefore also describes the probability of obtaining exactly correct balls in a pick- lottery from a reservoir of balls (of which n=N are "good" and m=r-N are "bad"). For example, for N=6 and r=36 , the probabilities of obtaining correct balls are given in the following table.

number correct	probability	odds
0	0.3048	2.280:1
1	0.4390	1.278:1
2	0.2110	3.738:1
3	0.04169	22.99:1
4	0.003350	297.5:1
5		10820:1
6

The th selection has an equal likelihood of being in any trial, so the fraction of acceptable selections is

(5)

i.e.,

(6)

The expectation value of is therefore simply

			(7)
			(8)
			(9)
			(10)

This can also be computed by direct summation as

			(11)
			(12)

The variance is

(13)

Since x_i is a Bernoulli variable,

			(14)
			(15)
			(16)
			(17)
			(18)

(19)

For i<j , the covariance is

(20)

The probability that both and are successful for i!=j is

			(21)
			(22)
			(23)

But since x_i and x_j are random Bernoulli variables (each 0 or 1), their product is also a Bernoulli variable. In order for x_ix_j to be 1, both x_i and x_j must be 1,

			(24)
			(25)
			(26)

Combining (26) with

			(27)
			(28)

gives

			(29)
			(30)

There are a total of N^2 terms in a double summation over . However, i=j for of these, so there are a total of N^2-N=N(N-1) terms in the covariance summation

(31)

Combining equations (◇), (◇), (◇), and (◇) gives the variance

			(32)
			(33)

so the final result is

(34)

and, since

(35)

and

(36)

we have

			(37)
			(38)
			(39)

This can also be computed directly from the sum

			(40)
			(41)

The skewness is

			(42)
			(43)

and the kurtosis excess is given by a complicated expression.

The generating function is

phi(t)=((m; N))/((n+m; N))_2F_1(-N,-n;m-N+1;e^(it)),

(44)

where _2F_1(a,b;c;z) is the hypergeometric function.

If the hypergeometric distribution is written

(45)

then

(46)

where is a constant.

REFERENCES:

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 532-533, 1987.

Feller, W. "The Hypergeometric Series." §2.6 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 41-45, 1968.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 113-114, 1992.

تصفح اقسام الموقع

Hypergeometric Distribution