- #1

psie

- 214

- 26

- Homework Statement
- One hundred numbers, uniformly distributed in the interval ##(0,1)## are generated by a computer. What is the probability that the largest number is at most ##0.9##? What is the probability that the second smallest number is at least ##0.002##?

- Relevant Equations
- The extreme order variables ##X_{(1)}=\min\{X_1,\ldots,X_{100}\}## and ##X_{(100)}=\max\{X_1,\ldots,X_{100}\}## have cdf ##F_{X_{(1)}}(x)=1-(1-F(x))^{100}## and ##F_{X_{(100)}}(x)=(F(x))^{100}## respectively, where ##F## is the cdf of the iid ##X_1,\ldots,X_{100}##.

The first question is fairly straightforward. The density of ##X## (i.e. one of the iid r.v.s. ##X_1,\ldots,X_{100}##) is just ##f(x)=1## for ##0<x<1## and ##0## otherwise. The cdf ##F## is therefore ##F(x)=x## for ##0<x<1##, ##F(x)=0## for ##x<0## and ##F(x)=1## for ##x>1##. In the first question, we are interested in \begin{align*} P(X_{(100)}<0.9)&=F_{X_{(100)}}(0.9) \\ &=(0.9)^{100}.\end{align*}For the 2nd question, I don't know how to approach this and I'm stuck. The cdf of arbitrary order variables has not been derived yet. I know we are looking for ##P(X_{(2)}>0.002)=1-P(X_{(2)}<0.002)##. But maybe there's a workaround. I only know the formulas in the relevant equations above. Grateful for any help.