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

Question

$X_1$, $X_2$, $X_3$ 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.

Answer 1

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 $1/\lambda$.

Answer 2

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.

Answer 3

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.