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If in an experiment I have recorded the number of people, lets say $X$, alive at some time ($>0$), out of a sample of $n$ people, which is the best distribution for $X$ to use?

The survival time of a single person is modelled as an exponential distribution with parameter θ: f(t)=θ*exp(-θt)

Since it is people it should be a discrete distribution and thus i am between poisson and binomial but I cannot decide which to use.

rogerdom
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  • What is the population from which the sample is taken, if you are counting how many in your sample are alive? – Justpassingby Jan 15 '16 at 20:02
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    Everything depends upon your assumed model. If you think each person has a probability $p$ of dying in a given year, then use an exponential. Under different assumptions, a Poisson, or binomial, or other is appropriate. Clarify your model first! – David G. Stork Jan 15 '16 at 20:05
  • It just states that the survival time of a single person is modelled as an exponential distribution with parameter theta. Apologies i forgot to mention this in the original post – rogerdom Jan 15 '16 at 20:07
  • I think you should decide whether to treat time as discrete or continuous. For the former, if you regard each subject as a geometrical r.v. with $p$ (whether dead or alive at each $t_i$), then $NB(n,p)$ (assuming the people are i.i.d) may serve your purpose as well. For continuous time, modelling each subject as exponential r.v. seems legit as well. – V. Vancak Jan 15 '16 at 20:15
  • Updated the question. I wrote it a bit rushed before, hope that it is clearer now – rogerdom Jan 15 '16 at 20:22

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"exponential distribution with parameter $\theta$" sometimes means an exponential distribution with expected value $\theta$, so that if $T$ is a random variable with such a distribution, then $\Pr(T>t) = e^{-t/\theta}$ for $t>0$, and sometimes means an exponential distribution with expected value $1/\theta$, so that $\Pr(T>t) = e^{-\theta t}$ for $t>0$. If $t$ is the particular time you have in mind, let $p=\text{either } e^{-t/\theta} \text{ or } e^{-\theta t}$ as the case may be. Then you're talking about the random number of successes in a fixed number $n$ of independent trials, with probability $p$ of success on each trial. Thus it has a binomial distribution.

Michael Hardy
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