How to calculate mortality rate or probability of death at time t for various parametric distributions, e.g. Weibull, exponential, log-normal

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!

• 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?) May 31, 2018 at 22:43
• 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. May 31, 2018 at 23:19
• It would be useful to identify which one (as well as following the requirement of offering credit) May 31, 2018 at 23:27
• You can also see this answer for general relationships between survival model functions Jun 16, 2022 at 11:55

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)