博文
The Hypergeometric Probability Distribution(超几何分布)
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1. A population contains a finite number $N$ of elements that possess one of two characteristics, say, red and black.
2. $r$ of the elements are red and the others black.
a1 = {Arrowheads[0.02], Arrow[{{10, 0.25}, {5, PDF[A[10], 5]}}]};
t1 = Text[Style["n=" <> ToString[10] <> ",r=50,N=100", Medium], {10, 0.25}, {-1, 0}];
a2 = {Arrowheads[0.02], Arrow[{{12, 0.23}, {10, PDF[A[20], 10]}}]};
t2 = Text[Style["n=" <> ToString[20] <> ",r=50,N=100", Medium], {12, 0.23}, {-1, 0}];
a3 = {Arrowheads[0.02], Arrow[{{25, 0.20}, {25, PDF[A[50], 25]}}]};
t3 = Text[Style["n=" <> ToString[50] <> ",r=50,N=100", Medium], {25, 0.20}, {0, -1}];
epilog = {a1, t1, a2, t2, a3, t3};
DiscretePlot[Evaluate@Table[PDF[A[n], k], {n, {10, 20, 50}}], {k, 0, 32},
PlotRange -> All, PlotMarkers -> Automatic, Epilog -> epilog,
Background -> RGBColor[0.1, 0.45, 0.3, 0.9]]
Expection and Variance:
If Y is a random variable with a hypergeometric distribution,then,
$$E(Y) = frac{{nr}}{N};;;{rm{and}};;;V(Y) = nleft( {frac{r}{N}} right)left( {frac{{N - r}}{N}} right)left( {frac{{N - n}}{{N - 1}}} right).$$
You can use the following Mathematica command to obtain these results
Expectation[x, x [Distributed] HypergeometricDistribution[n, r, N]]
or Mean[HypergeometricDistribution[n, r, N]]
Variance[HypergeometricDistribution[n, r, N]]
Property:
For a fixed fraction $p = frac{r}{N}$, the hypergeometric probability function converges to the binomial probability function as $N$ becomes large and $n$ is relatively small.
$$mathop {lim }limits_{N to infty } frac{{left( {begin{array}{*{20}{c}} r\ y end{array}} right)left( {begin{array}{*{20}{c}} {N - r}\ {N - y} end{array}} right)}}{{left( {begin{array}{*{20}{c}} N\ n end{array}} right)}} = left( {begin{array}{*{20}{c}} n\ y end{array}} right){p^y}{left( {1 - p} right)^{n - y}}.$$
https://blog.sciencenet.cn/blog-531885-597654.html
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