Question

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}\) form a random sample from the uniform distribution on the interval [0, 1]. Let \({{\bf{Y}}_{\bf{1}}}{\bf{ = min}}\left\{ {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}} \right\}\), \({{\bf{Y}}_{\bf{n}}}{\bf{ = max}}\left\{ {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}} \right\}\)and \({\bf{W = }}{{\bf{Y}}_{\bf{n}}}{\bf{ - }}{{\bf{Y}}_{\bf{1}}}\). Show that each of the random variables \({{\bf{Y}}_{\bf{1}}}{\bf{,}}{{\bf{Y}}_{\bf{n}}}\,\,{\bf{and}}\,\,{\bf{W}}\) has a beta distribution.

Short answer

It is proved that,

\({Y_1} \sim Beta\left( {1,n} \right)\),

\({Y_n} \sim Beta\left( {n,1} \right)\),

\(W \sim Beta\left( {n - 1,2} \right)\)

Step-by-step solution

Given information

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

\(\begin{array}{l}{Y_1} = \min \left\{ {{X_1} \ldots {X_n}} \right\}\\{Y_n} = \max \left\{ {{X_1} \ldots {X_n}} \right\}\end{array}\)

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

Therefore, the PDF of X is expressed as,

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

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

\(\begin{array}{c}{F_X}\left( x \right) = P\left( {X \le x} \right)\\ = \frac{{x - a}}{{b - a}}\\ = \frac{{x - 0}}{{1 - 0}}\\ = x\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ldots \left( 2 \right)\end{array}\)

The PDF of beta distribution with parameters\({\bf{\alpha }}\,\,{\bf{and}}\,\,{\bf{\beta }}\)is,

\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\frac{{{\bf{\Gamma }}\left( {{\bf{\alpha + \beta }}} \right)}}{{{\bf{\Gamma }}\left( {\bf{\alpha }} \right){\bf{\Gamma }}\left( {\bf{\beta }} \right)}}{{\bf{x}}^{{\bf{\alpha - 1}}}}{\left( {{\bf{1 - x}}} \right)^{{\bf{\beta - 1}}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\bf{0 < x < 1}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ldots \left( 3 \right)\)

Define the result

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, pdf of r-th order statistic, \({X_{\left( r \right)}}\)is given as:

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

Obtain the pdf of \({{\bf{Y}}_{\bf{1}}}\)

From equation (1) and (2), following this, the pdf of\({{\bf{Y}}_{\bf{1}}}\)is,

\(\begin{array}{c}g\left( {{y_1}} \right) = {}^n{C_1}F{\left( {{y_1}} \right)^{1 - 1}} \times f\left( y \right) \times {\left( {1 - F\left( y \right)} \right)^{n - 1}}\\ = \left\{ \begin{array}{l}n{\left( {1 - y} \right)^{n - 1}}\;\;\;\;\;\;\;\;0 < y < 1\\0\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{array} \right.\end{array}\)

Consider \(\alpha = 1,\beta = n\) in equation (3), this is the form Beta distribution.

Thus, \({Y_1} \sim Beta\left( {1,n} \right)\).

Obtain the pdf of \({{\bf{Y}}_{\bf{n}}}\)

From the equation above, the pdf of \({Y_{\left( n \right)}}\) is,

\(\begin{array}{c}g\left( {{y_n}} \right) = {}^n{C_n}F{\left( y \right)^{n - 1}} \times f\left( y \right) \times {\left( {1 - F\left( y \right)} \right)^{n - n}}\\ = \left\{ \begin{array}{l}n{y^{n - 1}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 < y < 1\\0\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{array} \right.\end{array}\)

Consider \(\alpha = n,\beta = 1\) in equation (3). This is the form Beta distribution.

Thus, \({Y_n} \sim Beta\left( {n,1} \right)\)

Obtain the P.D.F of   \({{\bf{Y}}_{\bf{n}}} - {{\bf{Y}}_{\bf{1}}}\)random variable

Define the random variable \(W = {Y_n} - {Y_1}\) .

The P.D.F for w,

\(\begin{aligned}{}{g_w}\left( y \right) &= n\left( {n - 1} \right){\left( {F\left( {y + w} \right) - F\left( y \right)} \right)^{n - 2}}f\left( {x + w} \right)\\ &= n\left( {n - 1} \right){\left( {y + w - y} \right)^{n - 2}}\left( {1 - w} \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;...\left[ {F\left( x \right) = x} \right]\\ &= n\left( {n - 1} \right){\left( w \right)^{n - 2}}\left( {1 - w} \right)\end{aligned}\)

Compare with the beta distribution for \(\alpha = n - 1\) and \(\beta = 2\). Thus, \(W \sim Beta\left( {n - 1,2} \right)\).