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(what happens to the equation number?)
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You mean, how to prove that $S$ is dense in $(0.1,1)$, right? Well, on the page "[[Distribution of prime numbers]]", in Section 6 "The difference between prime numbers", we have $ d_n \ll p_n^\delta $, where $p_n$ is the $n$-th prime number, and $ d_n = p_{n+1}-p_n $ is the difference between adjacent prime numbers; this relation holds for all $ \delta > \frac{7}{12} $; in particular, taking $ \delta = 1 $ we get $ d_n \ll p_n $, that is, $ \frac{d_n}{p_n} \to 0 $ (as $ n \to \infty $), or equivalently, $ \frac{p_{n+1}}{p_n} \to 1 $. Now, your set $S$ consists of numbers $ s_n = 10^{-k} p_n $ for all $k$ and $n$ such that $ 10^{k-1} < p_n < 10^k $. Assume that $S$ is not dense in $(0.1,1).$ Take $a$ and $b$ such that $ 0.1 < a < b < 1 $ and $ s_n \notin (a,b) $ for all $n$; that is, no $p_n$ belongs to the set
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{{MSC|62E}}
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\[
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{{TEX|done}}
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X = (10a,10b) \cup (100a,100b) \cup (1000a,1000b) \cup \dots \, ;
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\]
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A
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all $ p_n $ belong to its complement
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[[Probability distribution|probability distribution]] of a random variable $X$ which takes non-negative integer values, defined by the formula
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\[
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\begin{equation}
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Y = (0,\infty) \setminus X = (0,10a] \cup [10b,100a] \cup [100b,1000a] \cup \dots
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P(X=k)=\frac{ {k+m-1 \choose k}{N-m-k \choose M-m} } { {N \choose M} } \tag{*}
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\]
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\end{equation}
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Using the relation $ \frac{p_{n+1}}{p_n} \to 1 $ we take $N$ such that $ \frac{p_{n+1}}{p_n} < \frac b a $ for all $n>N$. Now, all numbers $p_n$ for $n>N$ must belong to a single interval $ [10^{k-1} b, 10^k a] $, since it cannot happen that $ p_n \le 10^k a $ and $ p_{n+1} \ge 10^k b $ (and $n>N$). We get a contradiction: $ p_n \to \infty $ but $ p_n \le 10^k a $.
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where the parameters <math>N,M,m</math> are non-negative integers which satisfy the condition <math>m\leq M\leq N</math>. A negative hypergeometric distribution often arises in a scheme of sampling without replacement. If in the total population of size <math>N</math>, there are <math>M</math>  "marked"  and <math>N-M</math>  "unmarked"  elements, and if the sampling (without replacement) is performed until the number of  "marked"  elements reaches a fixed number <math>m</math>, then the random variable <math>X</math> — the number of  "unmarked"  elements in the sample — has a negative hypergeometric distribution \eqref{*}. The random variable <math>X+m</math> — the size of the sample — also has a negative hypergeometric distribution. The distribution \eqref{*} is called a negative hypergeometric distribution by analogy with the
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[[Negative binomial distribution|negative binomial distribution]], which arises in the same way for sampling with replacement.
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The mathematical expectation and variance of a negative hypergeometric distribution are, respectively, equal to
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\begin{equation}
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m\frac{N-M} {M+1}
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\end{equation}
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and
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\begin{equation}
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m\frac{(N+1)(N-M)} {(M+1)(M+2)}\Big(1-\frac{m}{M+1}\Big) \, .
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\end{equation}
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When <math>N, M, N-M \to \infty</math> such that <math>M/N\to p</math>, the negative hypergeometric distribution tends to the
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[[negative binomial distribution]] with parameters <math>m</math> and <math>p</math>.
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The distribution function <math>F(n)</math> of the negative hypergeometric function with parameters <math>N,M,m</math> is related to the
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[[Hypergeometric distribution|hypergeometric distribution]] <math>G(m)</math> with parameters <math>N,M,n</math> by the relation
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\begin{equation}
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F(n) = 1-G(m-1) \, .
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\end{equation}
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This means that in solving problems in mathematical statistics related to negative hypergeometric distributions, tables of hypergeometric distributions can be used. The negative hypergeometric distribution is used, for example, in
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[[Statistical quality control|statistical quality control]].
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====References====
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{|
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|-
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|valign="top"|{{Ref|Be}}||valign="top"|  Y.K. Belyaev,  "Probability methods of sampling control", Moscow  (1975) (In Russian) {{MR|0428663}} 
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|-
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|valign="top"|{{Ref|BoSm}}||valign="top"|  L.N. Bol'shev,  N.V. Smirnov,  "Tables of mathematical statistics", ''Libr. math. tables'', '''46''', Nauka  (1983)  (In Russian)  (Processed by L.S. Bark and E.S. Kedrova) {{MR|0243650}} {{ZBL|0529.62099}}
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|-
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|valign="top"|{{Ref|JoKo}}||valign="top"|  N.L. Johnson,  S. Kotz,  "Distributions in statistics, discrete distributions", Wiley  (1969) {{MR|0268996}} {{ZBL|0292.62009}}
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|-
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|valign="top"|{{Ref|PaJo}}||valign="top"|  G.P. Patil,   S.W. Joshi,   "A dictionary and bibliography of discrete distributions", Hafner  (1968) {{MR|0282770}}
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|-
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|}

Revision as of 14:42, 5 June 2017

2020 Mathematics Subject Classification: Primary: 62E [MSN][ZBL]


A probability distribution of a random variable $X$ which takes non-negative integer values, defined by the formula \begin{equation} P(X=k)=\frac{ {k+m-1 \choose k}{N-m-k \choose M-m} } { {N \choose M} } \tag{*} \end{equation} where the parameters \(N,M,m\) are non-negative integers which satisfy the condition \(m\leq M\leq N\). A negative hypergeometric distribution often arises in a scheme of sampling without replacement. If in the total population of size \(N\), there are \(M\) "marked" and \(N-M\) "unmarked" elements, and if the sampling (without replacement) is performed until the number of "marked" elements reaches a fixed number \(m\), then the random variable \(X\) — the number of "unmarked" elements in the sample — has a negative hypergeometric distribution \eqref{*}. The random variable \(X+m\) — the size of the sample — also has a negative hypergeometric distribution. The distribution \eqref{*} is called a negative hypergeometric distribution by analogy with the negative binomial distribution, which arises in the same way for sampling with replacement.

The mathematical expectation and variance of a negative hypergeometric distribution are, respectively, equal to

\begin{equation} m\frac{N-M} {M+1} \end{equation}

and

\begin{equation} m\frac{(N+1)(N-M)} {(M+1)(M+2)}\Big(1-\frac{m}{M+1}\Big) \, . \end{equation}

When \(N, M, N-M \to \infty\) such that \(M/N\to p\), the negative hypergeometric distribution tends to the negative binomial distribution with parameters \(m\) and \(p\).

The distribution function \(F(n)\) of the negative hypergeometric function with parameters \(N,M,m\) is related to the hypergeometric distribution \(G(m)\) with parameters \(N,M,n\) by the relation \begin{equation} F(n) = 1-G(m-1) \, . \end{equation} This means that in solving problems in mathematical statistics related to negative hypergeometric distributions, tables of hypergeometric distributions can be used. The negative hypergeometric distribution is used, for example, in statistical quality control.

References

[Be] Y.K. Belyaev, "Probability methods of sampling control", Moscow (1975) (In Russian) MR0428663
[BoSm] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics", Libr. math. tables, 46, Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) MR0243650 Zbl 0529.62099
[JoKo] N.L. Johnson, S. Kotz, "Distributions in statistics, discrete distributions", Wiley (1969) MR0268996 Zbl 0292.62009
[PaJo] G.P. Patil, S.W. Joshi, "A dictionary and bibliography of discrete distributions", Hafner (1968) MR0282770
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Boris Tsirelson/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox&oldid=41610
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