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  "body": "\n## Definition\n\nLet $T$ be a random variable with values in $[0, \\infty)$ that has a [[probability density function|0ZY9DX8FBF]] $f$, and let $S$ be its [[survival function|QTM1HZM53Q]].\nThe *hazard function* of $T$ is defined for all $t \\ge 0$ with $S(t) \u003e 0$ by\n$$h(t) = \\frac{f(t)}{S(t)}.$$\nThe *cumulative hazard function* is $H(t) = \\int_0^t h(u) \\, du$.\n\n## Remarks\n\nAt every point $t$ where $f$ is continuous, the hazard is the limit of a [[conditional probability|0990W5PN13]] per unit time:\n$$h(t) = \\lim_{\\Delta \\downarrow 0} \\frac{P(t \\le T \u003c t + \\Delta \\mid T \\ge t)}{\\Delta}.$$\nSince $h = -S'/S = -(\\log S)'$ there, one gets $S(t) = e^{-H(t)}$, so $h$ determines the distribution of $T$.\n\nExample: for the [[exponential distribution|0E99F51GVB]] with rate $\\lambda$, $f(t) = \\lambda e^{-\\lambda t}$ and $S(t) = e^{-\\lambda t}$, so $h(t) = \\lambda$ is constant and $H(t) = \\lambda t$.\nFor the Weibull distribution with shape $k \u003e 0$ and scale $\\lambda \u003e 0$, $S(t) = e^{-(t/\\lambda)^k}$ and $h(t) = (k/\\lambda)(t/\\lambda)^{k-1}$, which increases when $k \u003e 1$ and decreases when $k \u003c 1$.\n\nNon-example: if $T$ takes only the values $1$ and $2$, each with probability $1/2$, then $T$ has no density and the formula does not apply.\nFor discrete times one uses instead the discrete hazard $P(T = t \\mid T \\ge t)$, here $1/2$ at $t = 1$ and $1$ at $t = 2$.\n",
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