What Is the Minimum Sample Size n for Pr[Yn≥0.99] to Be At Least 0.95?

[uva123]

Homework Statement

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

Homework Equations

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

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]

 

[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 $Y_n$