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  "body": "\n## Definition\n\nLet $X$ be a real random variable and let $k \\ge 1$ be an integer with $E[|X|^k] \u003c \\infty$. Let $\\mu = E[X]$.\nThe $k$-th central moment of $X$ is\n$$\\mu_k = E\\left[(X - \\mu)^k\\right].$$\n\n## Remarks\n\nAlways $\\mu_1 = 0$, and $\\mu_2 = \\operatorname{Var}(X)$ is the [[variance|007QED3E0Y]].\nCentral moments do not change when one adds a constant to $X$. For $a \\in \\mathbb{R}$, the central moments of $aX$ are $a^k \\mu_k$.\nThe binomial theorem gives $\\mu_k = \\sum_{j=0}^{k} \\binom{k}{j} (-\\mu)^{k - j} m_j$ in terms of the [[moments|0PH0VHTQS0]] $m_j = E[X^j]$, with $m_0 = 1$.\nFor a distribution that is symmetric about its mean, all odd central moments are $0$.\n\n## Explanations\n\n### L1 human\n\nCentral moments measure the shape of a distribution around its center, and not its position.\nThe third gives asymmetry and the fourth gives the weight of the tails.\n",
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