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Caleb Burke
2026-06-15 00:32:30 -07:00
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reading/000_Introduction to Probability, Statistics, and Random Processes/ch3/notes.ipynb
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"id": "2e1643c0",
"metadata": {},
"source": [
"***Definition.*** A random variable $X$ is said to be a *Hypergeometric* random variable with parameter $b$, $r$ and $k$, shown as $X \\sim \\text{Hypergeometric}(b,r,k),$ if its range is $R_X = \\{\\text{max}(0,k-r),\\text{max}(0,k-r)+1,\\text{max}(0,k-r)+2,\\text{min}(k,b)\\},$ and its PMF is given by\n",
"***Definition.*** A random variable $X$ is said to be a *Hypergeometric* random variable with parameter $b$, $r$ and $k$, shown as $X \\sim \\text{Hypergeometric}(b,r,k),$ if its range is $R_X = \\{\\text{max}(0,k-r),\\text{max}(0,k-r)+1,\\text{max}(0,k-r)+2,...,\\text{min}(k,b)\\},$ and its PMF is given by\n",
"\n",
"$$P_X(x) = \\begin{cases} \n",
"\\frac{\\binom{b}{x}\\binom{r}{k-x}}{\\binom{b+r}{k}} & \\text{for } x \\in R_X \\\\ \n",