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  "body": "\n## Definition\n\nLet $X$ be data whose law $P_\\theta$ depends on $\\theta \\in \\Theta$, and consider a [[hypothesis test|087P03NEND]] of $H_0 \\colon \\theta \\in \\Theta_0$ against $H_1 \\colon \\theta \\in \\Theta_1$.\nLet the test reject $H_0$ when $X$ lies in a rejection region $R$.\nThe *power function* of the test is $\\pi(\\theta) = P_\\theta(X \\in R)$ for $\\theta \\in \\Theta$.\nThe *power* of the test at $\\theta \\in \\Theta_1$ is $\\pi(\\theta)$.\n\n## Remarks\n\nFor $\\theta \\in \\Theta_1$, the power is $1 - \\beta(\\theta)$, where $\\beta(\\theta)$ is the probability of a [[type II error|0R41PVFA39]].\nFor $\\theta \\in \\Theta_0$, the value $\\pi(\\theta)$ is the probability of a type I error, and $\\sup_{\\theta \\in \\Theta_0} \\pi(\\theta)$ is the size of the test.\nFor a randomized test $\\varphi$ with values in $[0, 1]$, the power function is $\\pi(\\theta) = E_\\theta[\\varphi(X)]$.\nExample: let $X \\sim \\mathcal{N}(\\theta, 1)$, $H_0 \\colon \\theta = 0$, $H_1 \\colon \\theta \u003e 0$, and $R = \\set{x \u003e 1.645}$.\nThen $\\pi(\\theta) = 1 - \\Phi(1.645 - \\theta)$, where $\\Phi$ is the standard normal distribution function; so $\\pi(0) \\approx 0.05$ and $\\pi(2) \\approx 0.64$.\nNon-example: $\\pi(0) \\approx 0.05$ is not a power in this example, since $0 \\notin \\Theta_1$; it is the size.\n",
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