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  "body": "\n## Statement\n\nLet $p \\in (0, 1)$ and let $S_n \\sim \\operatorname{Bin}(n, p)$ for each $n \\ge 1$. Let $q = 1 - p$.\nThen $(S_n - np)/\\sqrt{npq}$ converges [[in distribution|0JDAQHWA9N]] to $\\mathcal{N}(0, 1)$. That is, for all reals $a \u003c b$,\n$$\\lim_{n \\to \\infty} P\\left(a \\le \\frac{S_n - np}{\\sqrt{npq}} \\le b\\right) = \\int_a^b \\frac{1}{\\sqrt{2\\pi}} e^{-x^2/2} \\, dx.$$\n\n## Remarks\n\nThis is the special case of the [[central limit theorem|0EPZC4Z1FC]] for Bernoulli variables.\nIndeed, $S_n$ has the law of a sum of $n$ independent Bernoulli($p$) variables.\nA local form also holds: $\\binom{n}{k} p^k q^{n-k} \\sim \\frac{1}{\\sqrt{2\\pi npq}} e^{-(k - np)^2/(2npq)}$ as $n \\to \\infty$, uniformly for $k$ with $|k - np| \\le C\\sqrt{n}$. The proof uses Stirling's formula.\nIn practice, one often adds a continuity correction of $1/2$ to the integer bounds.\n\n## History\n\nAbraham de Moivre proved the case $p = 1/2$. Pierre-Simon Laplace extended it to general $p$.\n\n## Explanations\n\n### L1 human\n\nFor many trials, the binomial distribution looks like a bell curve with mean $np$ and variance $np(1 - p)$.\nThis lets you approximate binomial probabilities, such as \"at least $520$ heads in $1000$ tosses\", with the normal distribution.\n",
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