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

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Homework Statement



Suppose that X1,...,Xn form a random sample from a inform 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

Homework Equations



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


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]
 
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Discussion

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

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

b) The individual [itex]X_i[/itex] values are independent

This should allow you to get the CDF of [itex]Y_n[/itex]