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  "body": "\n## Definition\n\nLet $X$ be a real random variable with $E[X^4] \u003c \\infty$, mean $\\mu$ and standard deviation $\\sigma \u003e 0$.\nThe kurtosis of $X$ is\n$$\\operatorname{Kurt}(X) = E\\left[\\left(\\frac{X - \\mu}{\\sigma}\\right)^4\\right] = \\frac{\\mu_4}{\\sigma^4},$$\nwhere $\\mu_4$ is the fourth [[central moment|0ZT24EC90K]]. The excess kurtosis is $\\operatorname{Kurt}(X) - 3$.\n\n## Remarks\n\nEvery [[normal distribution|07RXBVTVZB]] has kurtosis $3$, so its excess kurtosis is $0$.\nFor every $X$, $\\operatorname{Kurt}(X) \\ge 1$. The Bernoulli distribution with $p = 1/2$ reaches this bound.\nDistributions with positive excess kurtosis are called leptokurtic. Those with negative excess kurtosis are called platykurtic.\nThe kurtosis does not change under $X \\mapsto aX + b$ with $a \\ne 0$.\n\n## Explanations\n\n### L1 human\n\nKurtosis compares the tails of a distribution with those of the normal distribution.\nA high kurtosis means that rare extreme values are more frequent than for a normal variable with the same variance.\n",
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