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Chapter 5: Q14E (page 345)

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

Expert verified

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

01

Given information

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

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

02

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,

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

03

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:

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

04

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

05

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

06

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

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Most popular questions from this chapter

Suppose that the random variables \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{k}}}\) are independent and that \({{\bf{X}}_{\bf{i}}}\) has the Poisson distribution with mean \({{\bf{\lambda }}_{\bf{i}}}\left( {{\bf{i = 1, \ldots ,k}}} \right)\). Show that for each fixed positive integer n, the conditional distribution of the random Vector \({\bf{X = }}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{k}}}} \right)\), given that \(\sum\limits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{X}}_{\bf{i}}}{\bf{ = n}}} \) it is the multinomial distribution with parameters n and

\(\begin{array}{l}{\bf{p = }}\left( {{{\bf{p}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{p}}_{\bf{k}}}} \right){\bf{,}}\,{\bf{where}}\\{{\bf{p}}_{\bf{i}}}{\bf{ = }}\frac{{{{\bf{\lambda }}_{\bf{i}}}}}{{\sum\limits_{{\bf{j = 1}}}^{\bf{k}} {{{\bf{\lambda }}_{\bf{j}}}} }}\,{\bf{for}}\,{\bf{i = 1, \ldots ,k}}{\bf{.}}\end{array}\)

Suppose that in a large lot containingTmanufactured items, 30 percent of the items are defective, and 70 percent are non-defective. Also, suppose that ten items are selected randomly without replacement from the lot.

Determine (a) an exact expression for the probability that not more than one defective item will be obtained and (b) an approximate expression for this probability based on the binomial distribution.

Suppose that two players A and B are trying to throw a basketball through a hoop. The probability that player A will succeed on any given throw is p, and he throws until he has succeeded r times. The probability that player B will succeed on any given throw is mp, where m is a given integer (m = 2, 3, . . .) such that mp < 1, and she throws until she has succeeded mr times.

a. For which player is the expected number of throws smaller?

b. For which player is the variance of the number of throws smaller?

Find the 0.5, 0.25, 0.75, 0.1, and 0.9 quantiles of the standard normal distribution.

Suppose that the proportion X of defective items in a large lot is unknown and that X has the beta distribution with parameters\({\bf{\alpha }}\,\,{\bf{and}}\,\,{\bf{\beta }}\).

a. If one item is selected at random from the lot, what is the probability that it will be defective?

b. If two items are selected at random from the lot, what is the probability that both will be defective?
See all solutions

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