Minimum and Maximum as Random Variables If we have random variables $A,B,C$, how do we find the density and distribution functions for $\min{\left(A,B,C\right)}$, and $\max{\left(A,B,C\right)}$? I'm asking for the general case, (discrete/absolute continuous, independent/non-independent).
Perhaps, you can illustrate on something simple like $X,Y \sim UNI[0,1]$, for example?
 A: Illustration of one commonly used approach:
Minimum. Let $X_1 \sim \mathsf{Exp}(\lambda_1)$ and independently,
$X_2 \sim\mathsf{Exp}(\lambda_2).$ Then $V = \min(X_1,X_2)$ has reliability function
$$1 - F_V(v) = R_v(V > v) = P(X_1 > v, X_2 > v)\\ =
  P(X_1 > v)P(X_2 > v) = e^{-\lambda_1 v}e^{-\lambda_2 v}
= e^{-(\lambda_1+\lambda_2)v},$$
for $v > 0.$
This is the reliability function of $\mathsf{Exp}(\lambda_1+\lambda_2).$
The method extends to more than two random variables. The product is not always so easily summarized as in this example, but the approach is widely applicable.
Maximum. For $W=\max(X_1 + X_2).$ begin with
the CDF $$F_W(w) = P(W\le w) = P(X_1 < w,X_2 \le w).$$
In the case of two exponential random variables, the maximum is not another exponential random variable. But this method also extends to more than two random variables, and is widely applicable.
Simulation. Sometimes, especially for more difficult cases than exponential, it is helpful to have approximate descriptions of the distribution of minimum or maximum, which can often
be obtained easily by simulation. For example, for the minimum of independent $X_1 \sim \mathsf{Exp}(3)$
and $X_2 \sim \mathsf{Exp}(4),$ a simulation in R might
go as follows:
set.seed(2020)
x1 = rexp(10^6, 3);  x2 = rexp(10^6, 4)
v = pmin(x1, x2)
summary(v);  sd(v)
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
0.0000003 0.0412125 0.0990582 0.1429672 0.1981739 1.9674862 
[1] 0.1429503  # aprx SD(V) = 1/7 =  0.1428571

hist(v, prob=T, br=50, col="skyblue2", main="EXP(7)")
 curve(dexp(x, 7), add=T, col="orange", lwd=2)


