Question

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}\)form a random sample of nobservations from the uniform distribution on the interval(0, 1), and let Y denote the second largest of the observations.Determine the p.d.f. of Y.Hint: First, determine thec.d.f. G of Y by noting that

\(\begin{aligned}G\left( y \right) &= \Pr \left( {Y \le y} \right)\\ &= \Pr \left( {At\,\,least\,\,n - 1\,\,observations\,\, \le \,\,y} \right)\end{aligned}\)

Short answer

\(g\left( y \right) = \left\{ \begin{array}{l}n{x^{n - 2}}\left( {1 - x} \right),0 < y < 1\\0,\,otherwise\end{array} \right.\)

Step-by-step solution

Given information

The random variable X follows uniform distribution on the interval [0,1], i.e.,\(X \sim U\left[ {0,1} \right]\).

Here\({X_1} \ldots {X_n}\)is a random sample from a uniform distribution\(U\left( {0,1} \right)\).

\(Y = \sec ond\,\,\max \left\{ {{X_1} \ldots {X_n}} \right\}\)

Obtain the PDF and CDF of X

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

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

\({f_x} = \frac{1}{{1 - 0}};0 \le x \le 1\)

\(\begin{aligned}{f_x} &= 1,0 \le x \le 1\\ &= 0,o.w. \ldots \left( 1 \right)\end{aligned}\)

The CDF of a uniform distribution is obtained by using the formula:

\(\begin{aligned}{F_X}\left( x \right) &= P\left( {X \le x} \right)\\ &= \frac{{x - a}}{{b - a}}\\ &= \frac{{x - 0}}{{1 - 0}}\\ &= x\,{\rm{for}}\,\,0 < x < 1 \ldots \left( 2 \right)\end{aligned}\)

Obtain the pdf of the second largest order statistic

By following the result,

Let \({X_1} \ldots {X_n}\)be a random sample of size n from a population with pdf \(f\left( x \right)\)and cdf\(F\left( x \right)\). Then, a pdf of rth order statistic \({X_{\left( r \right)}}\)is given as:

\(g\left( x \right) = {}^n{C_r}F{\left( x \right)^{r - 1}} \times f\left( x \right) \times {\left( {1 - F\left( x \right)} \right)^{n - r}}\)

In our case \(f\left( x \right) = 1\) and \(F\left( x \right) = x\)from (1) and (2)

Following this, the pdf of \(Y = {X_{\left( {n - 1} \right)}}\) is

\(\begin{aligned}g\left( y \right) &= {}^n{C_{n - 1}}F{\left( x \right)^{n - 1 - 1}} \times f\left( x \right) \times {\left( {1 - F\left( x \right)} \right)^{n - \left( {n - 1} \right)}}\\ &= n{x^{n - 2}}\left( {1 - x} \right)\end{aligned}\)

Therefore, the pdf of Y is:

\(g\left( y \right) = \left\{ \begin{array}{l}n{x^{n - 2}}\left( {1 - x} \right),0 < y < 1\\0,\,otherwise\end{array} \right.\)