91Ó°ÊÓ

Suppose\({X_i}\)is the random sample that follows uniform distribution on the interval [0,1] that is \(X \sim U\left[ {0,1} \right]\).

Also, \({Y_{\left( n \right)}} = \max \left( {{X_1} \ldots {X_n}} \right)\)

The pdf of a uniform distribution is obtained by using the formula: \(\frac{1}{{b - a}};a \le x \le b\).

Here, \(a = 0,b = 1\).

Therefore, the PDF of X is expressed as,

\({f_x}\left( x \right) = \left\{ \begin{array}{l}\frac{1}{{1 - 0}} = 1\;\;\;\;\;\;\;\;\;\;0 \le x \le 1\\0;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{array} \right.\)

Define, \({Y_{\left( n \right)}} = \max \left( {{X_1} \ldots {X_n}} \right)\)

The CDF of \({Y_{\left( n \right)}}\) is,

\(\begin{aligned}P\left( {{Y_{\left( n \right)}} \le y} \right) &= P\left( {\max \left( {{X_1} \ldots {X_n}} \right) \le y} \right)\\ &= P\left( {{X_1} \le y} \right) \ldots P\left( {{X_n} \le y} \right)\;\left( {{\rm{independence}}} \right)\\ &= \left( {\int\limits_0^y {f\left( {{x_1}} \right)d{x_1}} } \right) \ldots \left( {\int\limits_0^y {f\left( {{x_n}} \right)d{x_n}} } \right)\\ &= \left( {\int\limits_0^y {1d{x_1}} } \right) \ldots \left( {\int\limits_0^y {1d{x_n}} } \right)\\ &= y \times \ldots \times y\\ &= {y^n}\end{aligned}\)

Taking derivative of cdf to obtain the pdf,

\(\begin{aligned}{f_{Y\left( n \right)}}\left( x \right) &= \frac{d}{{dx}}\left( {{y^n}} \right)\\ &= \left\{ \begin{aligned}n{y^{n - 1}}\,\;\;\;\;\;\;\;\;\;\;\;\;\;0 < {y_{\left( n \right)}} < 1\\0\;\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{aligned} \right.\end{aligned}\)

Now, \(\Pr \left( {{Y_n} \ge 0.99} \right) \ge 0.95\)

\(\begin{aligned}\int\limits_{0.99}^1 {n{{\left( y \right)}^{n - 1}}dy} \ge 0.95\\\left( {{y^n}} \right)_{0.99}^1 \ge 0.95\\{1^n} - {0.99^n} \ge 0.95\\{0.99^n} \le 0.05\end{aligned}\)

Apply log on both sides as,

\(\begin{aligned}n\log \left( {0.99} \right) &\le \log \left( {0.05} \right)\\n &\le \frac{{\log \left( {0.05} \right)}}{{\log \left( {0.99} \right)}}\\n &= 298.07\\n &\approx 298\end{aligned}\)

Hence, minimum value of n is 298.