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If I have $\lambda$ and $\gamma$, can I estimate that the probability of death at time $t$, based on a Weibull distribution is:

$$pDeath(t) = 1 - e^{\lambda(t_i^\gamma - t_{i+1}^\gamma)}$$

What are similar formulas for the exponential, Gompertz, log-normal, log-logistic, and logistic distributions?

If possible, please also provide the logic of how to derive the probability of death at time t.

Thank you!

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  • $\begingroup$ When you say "based on a Weibull distribution" - what is it the distribution of? (i.e. what does the variable represent? ... and where did you get the formula from?) $\endgroup$
    – Glen_b
    May 31, 2018 at 22:43
  • $\begingroup$ I found the p(death) formula on a website. The parametric distributions would be used to estimate survival curves, from which I'd want to estimate the probability of death at time t. Thank you again. $\endgroup$
    – justtrying
    May 31, 2018 at 23:19
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    $\begingroup$ It would be useful to identify which one (as well as following the requirement of offering credit) $\endgroup$
    – Glen_b
    May 31, 2018 at 23:27
  • $\begingroup$ You can also see this answer for general relationships between survival model functions $\endgroup$ Jun 16, 2022 at 11:55

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The survival function of a distribution is $S_X(x) = 1-F_X(x)$,where $F$ is the cdf. That is $S_X(x)$ is $P(X>x)$.

When you say "the parametric distributions would be used to estimate survival curves" you could think of it as$^\dagger$ estimating the parameters of a distribution by fitting $S(t)$ (continuous in all the above cases, and now using $t$ rather than $x$ because we're dealing with survival time) to the proportion of cases alive past time $t$, for each $t$ (which for a sample is a step function). That is your model is saying that for some individual $j$, $P(j \text{ survives past time } t) = S(t)$, where the function $S$ depends on some collection of parameters.

[The Weibull cdf (in the parameterization in your question) is $F_X(x) = 1 - e^{-\lambda x^\gamma}$, and $S_X(x)$ is correspondingly $e^{-\lambda x^\gamma}$.]

The formula you quote is for the probability that the individual dies between times $t_i$ and $t_{i+1}$ (i.e. $S(t_i)-S(t_{i+1})$, where $t_{i+1}>t_i$), not the probability that the individual dies at time $t$.

You can calculate the corresponding probability of death in the time between $t_i$ and $t_{i+1}$ for the other continuous distributions by either finding the survivor function $S=1-F$, and then writing $S(t_i)-S(t_{i+1})$ or equivalently directly in terms of $F$ as $F(t_{i+1})-F(t_i)$. Wikipedia lists the cdfs for most of these so it's merely a matter of substitution (some cannot be written in closed form, however).

Because the survival time is continuous, the probability of death at time $t$ is $0$. This will be the case for all the distributions you quote.

You could, however, consider the probability of death within a small interval of time after $t$, in which would approximately be the density times the width of the interval.

[When you say "mortality rate or probability of death" it's not 100% clear whether you are asking for the hazard function or that calculation.]

$\dagger$ (glossing over some important details)

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