# Help! Theoretical Stats

1. May 10, 2010

### uva123

1. The problem statement, all variables and given/known data

Suppose that X1,....,Xn form a random sample from a unifom distribution on the interval [0,1] and that the random variable Ynmax{X1,....,Xn}.
Determine the smallest value of n such that Pr[Yn≥.99]≥.95

2. Relevant equations

W=Yn-Y1 where W is the range of the sample
Y1=Z
Yn=W+Z
Y1=min{X1,...,Xn}

3. The attempt at a solution

f(x)= 1 for 0<x<1
F(x)=x for 0<x<1
h(w,z)=0 unless 0<w<1 and 0<z<1-w
G1(y)=Pr(1<y)= 1-Pr(Y1>y)= 1-[1-F(y)]n
=1-[1-x]

2. May 11, 2010

### statdad

Why are you considering the range? Your question concerns the maximum of the sample.

Two hints, using your notation of $Y_n$ as the maximum value.
a) If $Y_n \le y$, you know that $X_i \le y$ for $i = 1, 2, \dots, n$

b) The individual $X_i$ values are independent

This should allow you to get the CDF of $Y_n$

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