# How to calculate the distribution of the minimum of multiple exponential variables?

$X_1$, $X_2$, $X_3$ are independent random variables, each with an exponential distribution, but with means of $2.0, 5.0, 10.0$ respectively. Let $Y$= the smallest or minimum value of these three random variables. Derive and identify the distribution of $Y$. (The distribution function may be useful).

How do I solve this question? Do I plug in each mean to the exponential distribution? I would appreciate it if someone could explain this to me, thanks.

• Wikipedia has an easy answer :) Apr 10, 2013 at 22:02

When asked to derive the distribution of a random variable it's customary to present the cumulative distribution function (cdf), commonly denoted $F_Y(x):=\mathbb{P}(Y\leq x)$, for r.v. $Y$. In your case, it is probably helpful to note that $\mathbb{P}(Y\leq x)=1-\mathbb{P}(Y> x)$. Now, the minimum of 3 variables is of course greater than $x$ exactly when (iff) all of them are greater than $x$. You then get that $\mathbb{P}(Y> x)=\mathbb{P}(X_1>x, X_2>x,X_3>x)=\mathbb{P}(X_1>x)\mathbb{P}(X_2>x)\mathbb{P}(X_3 >x)$, where the last step follows from independence of the $\{X_i\}$.

Putting things together, $\mathbb{P}(Y\leq x)=1-\mathbb{P}(X_1>x)\mathbb{P}(X_2>x)\mathbb{P}(X_3 >x)$. Note now that $\mathbb{P}(X_i >x)=e^{-\lambda_ix},\forall i$ and you can probably fill in the last details yourself, i.e. simplify and note that $Y$ is also exponentially distributed and find its parameter.

As pointed out by @Drew75 in the comments, one should keep in mind that the mean of an exponential random variable with parameter $\lambda$ is equal to $1/\lambda$.

• Very clear explanation! You might add that in the case the mean of an exponential is equal to $\lambda^{-1}$. Jan 23, 2014 at 8:47
• Would you be able to get there via the union that if the $min(X_1, X_2, X_3) \leq x$, then one of the Variables has to be less than X and that is $P(X_1 \leq x)+P(X_2 \leq x)+P(X_3 \leq x)$ ? Edit: You can, I just figured it out. Inclusion, exclusion, but more work. Jul 25, 2021 at 7:42

The key (general) idea is that $Y=\min \{X_1,\dots,X_n\}> t$ if and only if each $X_i> t$. Using this and the independence assumption, you can compute $$F_Y(t) = P(Y\leq t)=1-P(Y>t) \, .$$ Do it yourself before looking at any available derivations. It is a simple and beautiful result.

You have an Exponential($\lambda$) parent where identicality is relaxed by replacing parameter $\lambda$ with $\lambda_i$ for i=1,…,3.

The pdf of the minimum order statistic (1st order statistic in a sample of size 3, with non-identical parameters) is given by the mathStatica function OrderStatNonIdentical:

For your parameter values, the pdf is simply:

All done.

• The question asks two things: derive and identify. Your software has done a beautiful job of the latter, but it reveals nothing about the derivation. Can it?
– whuber
Jun 26, 2013 at 13:24
• @whuber Can it or ... should it? As for the can it question, the software does have an option called VerboseMode[On] which shows all the integrals being performed in the background. This is done mostly to allow for external testing and verification purposes. In a complicated calculation, it would not show how all these integrals were being combined. Which then raises the second question: should it? Should it show HOW? I'm not sure that that is the appropriate role for software ... I suspect that role is much better (and already) served by many textbooks. Jun 26, 2013 at 17:16
• The other possibly relevant issue, of course, is whether it is always appropriate to answer the "How to do this step by step", especially if the question is actually someone's homework assignment. Computer software can help student's check that they have the correct solution ...and so have something to aim for ... but they still have to do the work/steps themselves (which is a plus, in my view). Jun 26, 2013 at 17:18
• I agree with your comments about appropriateness. Because this question asks how, though, it is begging for some guidance and insight into how one finds the result, if not a fully detailed derivation. Seeing that there is software which can produce the result does not satisfy that need. (In fact, that can even add to any frustration that the OP might be feeling.) In general, answers on this site that are software-only are valued less than answers that are accompanied with some explanation. The clearer the explanation (illustrations help immensely), the better.
– whuber
Jun 26, 2013 at 17:38
• True. Of course, the way we solve things .. the how ... changes with time. If someone asked: how to find the value of 23/7 to 8 decimal places, I would tell them to type it in a calculator. A few years before, someone might have suggested using a slide rule ... before then tables ... before then long division etc. For this particular problem, the how is already quite nicely solved by the link given above to wiki ... but that's not a very general solution because it is distribution specific. The modern general solution is to use a 'calculator' that can solve such problems generally. Jun 26, 2013 at 17:55