One of my textbooks lists the cumulative hazard function as H(t) = $\int_{0}^{t} h(u) \,du$.
Why are $u$ and $du$ used here instead of $t$ and $dt$?
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4https://math.stackexchange.com/questions/109105 – 311411 Sep 16 '21 at 00:06
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You can use $T$ if you don't like $u$, $$H(t) = \int_0^T h(T), dT$$ or any other letter than $t$. – Azlif Sep 16 '21 at 00:09
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@azlif one of your t's is wrong. – MJD Sep 16 '21 at 12:44
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The $u$ may be to avoid confusing you, or for the purpose of making the following discussion clearer. In $$\int_0^t h(u)\;du$$ it's clear that there are two unrelated variables, $t$ and $u$, with $t$ free and $u$ bound.
If it were written as $$\int_0^t h(t)\;dt$$ the meaning would be exactly the same, but now the two unrelated variables would have the same name, $t$. This could be confusing. If the text discussed the variable $t$ the reader might think it was talking about the bound variable $t$ when it actually meant the free variable $t$.
MJD
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