Let $X_1, X_2, X_3, \cdots,X_n$ be independent and identically distributred (iid) random variables. Then, how would you know/calculate what $min(X_1, X_2, X_3, \cdots,X_n)$ is?

  • $\begingroup$ How would you know or calculate the minimum of $n$ real numbers? The process is no different. See, for instance, my discussion of sums of random variables and just substitute "min" for "sum." $\endgroup$ – whuber Dec 12 '19 at 17:10
  • $\begingroup$ @whuber I was wondering since random variables don't have explicit variables and have random values. If a random variable would be smaller than another, would that mean all of its possible random values are less than the smallest possible value for the other random variable? $\endgroup$ – user12055579 Dec 12 '19 at 17:14
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    $\begingroup$ You need to study what a random variable is. $\endgroup$ – whuber Dec 12 '19 at 17:29

In a way you are quite right: You cannot calculate $min\{X_1,X_2\}$. Consider first the definition of a stochastic variable

Definition: A stochastic variable is a measurable function defined on a probability space whose outcomes are typically real numbers (wiki).

OK, so that means $X_1$ and $X_2$ are functions ...

But does that then mean you are supposed to read "$min\{X_1,X_2\}$" as a question about which of the functions are smaller than the other? This seems problematic because obviously most functions do not take on only a single value. As you point out in your comment

"would that mean all of its possible random values are less than the smallest possible value for the other random variable?" ...

You have the same challenge for the expression "$min\{x^2,y^2\}$". Are you supposed to compare the function $x^2$ to see if it is smaller or larger than the function $y^2$?

In both context the answer is NO. You are supposed to read $g(x,y)=min\{x^2,y^2\}$, not as a comparison between two functions but as a definition of a function $g(x,y)$ that for a given $x$ and $y$ compares the values $x^2$ and $y^2$ and returns the smallest two values.

What then happens when you insert random variables in the function? Well a function of random variables is itself a random variable. So $\min\{X_1,X_2\}$ is a definition of a random variable - we could write $Y = \min\{X_1,X_2\}$ and you are not supposed to calculate its value instead it takes random values as function of $X_1$ and $X_2$. And as soon as you know the realizations of $X_1$ and $X_2$ calculation is trivial, for example let $X_1 =1$ and $X_2=2$ then $Y=1$.

There is a somewhat related confusion displayed in this question conditional expectation. An expectation $\mathbb E[Z]$ is simply a constant $\mu$. The conditional expectation $\mathbb E[Z\lvert W=w]$ returns a constant for a given $w$ hence it is a function $g(w) = \mathbb E[Z\lvert W=w]$. And finally $g(W)$ is a random variable $\mathbb E[Z\lvert W]$ with the randomness originating from $W$. The point here is simply that remembering the definition of a stochastic variable and the fact that a function of stochastic variables is a stochastic variable is helpful (which shouldn't be a surprise).

  • $\begingroup$ Brilliant, once again! $\endgroup$ – user12055579 Dec 12 '19 at 23:02

I think the first order statistic probably the answer you want.

you can get the distribution function of $r$-th order statistic by this equation: $$ F_{X_(r)}(x) = \sum^n_{j=r} \binom{n}{j}[F_X(x)]^j[1-F_X(x)]^{n-j} $$ and r = 1 is the minimun of these random variables.

If you need more (like margin pdf), you can check on Wikipedia. There is some cases of specified distribution.

  • $\begingroup$ The question asks about how to define a function of a sequence of random variables. It does not ask about distributions. $\endgroup$ – whuber Dec 12 '19 at 19:19
  • $\begingroup$ Yes, this is where my question originated from. $\endgroup$ – user12055579 Dec 12 '19 at 23:03

This is more a comment than an answer but as a newbie I can't comment yet.

What do you mean by calculate?

  • If you mean retrieve the distribution, then without knowing the original distribution this will be hard.

  • If by calculate you mean in practice what does it mean, then it just means that you repeat your experiment $n$ times in order to get n values, you then take the minimum of those values. If you repeat this process a significant amount of time you will be able to plot the distribution of your new random variable.

To illustrate with an example, if n is large and X is evenly distributed on a finite interval [a,b], here your new r.v. will also be defined on [a,b] but the probability to get values around $a$ will be super high.

  • $\begingroup$ Yeah, I misunderstood the "calculate" part. The accepted answer makes me realize they are functions and so the answer wouldn't just be some random variable from $X_1, X_2, X_3, \cdots, X_n$. $\endgroup$ – user12055579 Dec 12 '19 at 23:05

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