TITLE: Minimum functions
QUESTION [1 upvotes]: Suppose that $X$ is uniformly distributed on the interval $[0,10]$ and $Y$ is uniformly distributed on the interval $[0.10]$ and that they are independent. 
Find the distribution function and the density function for $\min(X,Y).$ 
The book gives the formula for the min density function as $f_v(x)=n f_X(x)(1-F_X(x))^{n-1}$, however I have no idea what to plug in for $n.$ 
So far I have $f_v(x)=n \frac{1}{100}(1-\frac{x}{100})^{n-1}$ but this doesn't even help me get close to the answer in the back of the book which is $\frac{10-t}{50}.$
Can someone please help me apply this formula? Thanks in advance.

REPLY [1 votes]: One more thing: you need to say this is the density function for $0 < x < 10$, and that the density function is 0 otherwise.
Another method, in case it helps you: Let $V = \min(X, Y).$ Then
for $x \in (0, 1),$ we have
$$P(V > x) = P(X > x)P(Y > x) = [(10 - x)/10]^2.$$
so $F_V(x) = 1 - [(10 - x)/10]^2$ and $f_V(x) = F_V^\prime(x),$
which gives the answer in the back of the book.
Also, here is a brief simulation of 100,000 such minimums. At left, is a histogram of the results with an overlay of the density
function from the book. At right, is a plot (heavy black) of the empirical cumulative
distribution function (ECDF) with an overlay of the CDF (in green) found
above.
 m = 10^5
 x = runif(m, 0, 10);  y = runif(m, 0, 10)
 u = pmin(x, y)
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