Chapter 6: Q.6.28 (page 277)

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

Short Answer

Expert verified

$\frac{(2 n + 1)!}{n! n!} {(x)}^{n} {(1 - x)}^{n}$ is a Beta function with parameters $l = n + 1 & m = n + 1$ .

Step by step solution

Uniform distribution :

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

Explanation :

Sample size $2 n + 1$ .

Uniform distribution on $(0, 1)$ .

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

Sample size $2 n + 1$

Uniform distribution over $(0, 1)$

For median $j = n + 1$

Applying equation $(6.2)$

localid="1647349001706" $f_{x_{i j}} (x) = \frac{n!}{(n - j)! (j - 1)!} {[F (x)]}^{- 1} {[1 - F (x)]}^{n - j} f (x) f (x) = 1 \Rightarrow F (x) = x \Rightarrow f_{x_{(n +!)}} (x) = \frac{(2 n + 1)!}{n! n!} {(x)}^{n} {(1 - x)}^{n} \times 1 = \frac{(2 n + 1)!}{n! n!} {(x)}^{n} {(1 - x)}^{n} . . . . . . . . . (1)$

A form of Beta distribution is

localid="1648088016166" $f (x) = \frac{1}{β (l, m)} x^{l - 1} {(1 - x)}^{m - 1} 0 \leq x \leq 1; l, m > 0$

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

Hence proved.

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

Short Answer

Step by step solution

Uniform distribution :

Explanation :

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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

Step by step solution

Uniform distribution :

Explanation :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Statistics

Mechanics Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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