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Chapter 6: Q.6.28 (page 277)

Show that the median of a sample of size 2n+1from a uniform distribution on (0,1) has a beta distribution with parameters (n+1,n+1).

Short Answer

Expert verified

(2n+1)!n!n!(x)n(1-x)nis a Beta function with parametersl=n+1&m=n+1.

Step by step solution

01

Uniform distribution :

A continuous probability distribution is a Uniform distribution that describes occurrences that are equally likely to happen.

02

Explanation :

Sample size2n+1.

Uniform distribution on(0,1).

Beta distribution with parameters(n+1,n+1).

Sample size2n+1

Uniform distribution over(0,1)

For medianj=n+1

Applying equation (6.2)

localid="1647349001706" fxij(x)=n!(n-j)!(j-1)!F(x)-11-F(x)n-jf(x)f(x)=1鈬�F(x)=x鈬�fx(n+!)(x)=(2n+1)!n!n!(x)n(1-x)n脳1=(2n+1)!n!n!(x)n(1-x)n.........(1)

A form of Beta distribution is

localid="1648088016166" f(x)=1尾(l,m)xl-1(1-x)m-10鈮�x鈮�1;l,m>0

Thus, (1)is a Beta function with parameters l=n+1&m=n+1

Hence proved.

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Most popular questions from this chapter

If X and Y are jointly continuous with joint density function fX,Y(x, y), show that X + Y is continuous with density function fX+Y(t)=q鈭�qfX,Y(x,t鈭�x)dx

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A television store owner figures that 45 percent of the customers entering his store will purchase an ordinary television set, 15 percent will purchase a plasma television set, and 40 percent will just be browsing. If 5 customers enter his store on a given day, what is the probability that he will sell exactly 2 ordinary sets and 1 plasma set on that day?

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