negative hypergeometric random variable, example of

Suppose you have 7 black marbles and 10 white marbles in a jar. You pull marbles until you have 3 black marbles in your hand. $X$ would represent the number of white marbles in your hand.

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The expected value of $X$ would be $E[X]=\\frac{Wb}{B+1}=\\frac{3(10)}{7+1}=3.75$
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The variance of $X$ would be $Var[X]=\\frac{Wb(B-b+1)(W+B+1)}{(B+2)(B+1)^{2}}=\\frac{10(3)(7-3+1)(10+7+1)}{(7+%2)(7+1)^{2}}=1.875$
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The probability of having 3 white marbles would be $f_{X}(3)=\\frac{{3+b-1\\choose 3}{W+B-b-3\\choose W-3}}{{W+B\\choose W}}=\\frac{{3+%3-1\\choose 3}{10+7-3-3\\choose 10-3}}{{10+7\\choose 10}}=0.1697$

单词	NegativeHypergeometricRandomVariableExampleOf
释义	negative hypergeometric random variable, example of Suppose you have 7 black marbles and 10 white marbles in a jar. You pull marbles until you have 3 black marbles in your hand. $X$ would represent the number of white marbles in your hand. $\\star$ The expected value of $X$ would be $E[X]=\\frac{Wb}{B+1}=\\frac{3(10)}{7+1}=3.75$ $\\star$ The variance of $X$ would be $Var[X]=\\frac{Wb(B-b+1)(W+B+1)}{(B+2)(B+1)^{2}}=\\frac{10(3)(7-3+1)(10+7+1)}{(7+%2)(7+1)^{2}}=1.875$ $\\star$ The probability of having 3 white marbles would be $f_{X}(3)=\\frac{{3+b-1\\choose 3}{W+B-b-3\\choose W-3}}{{W+B\\choose W}}=\\frac{{3+%3-1\\choose 3}{10+7-3-3\\choose 10-3}}{{10+7\\choose 10}}=0.1697$
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