Chapter 6: Q.6.30 (page 277)

Compute the density of the range of a sample of size $n$ from a continuous distribution having density function $f$ .

Short Answer

Expert verified

Density of a sample of size $n$ is $P (R \leq a) = n \int_{-}^{\infty} {[F (x_{1} + a) - F (x_{1})]}^{- 1} f (x_{1}) d x_{1}$ .

Step by step solution

Joint probability distribution :

The related probability distribution on all possible pairings of outputs is the joint probability distribution. For each given number of random variables, the joint distribution may be studied.

Explanation :

Density function $f$ and sample size $n$ ,

$P \{R \leq a\} = P \{X_{(n)} - X_{(0)} \leq a\} = \iint_{x_{(i)} - x_{(i)}} \int_{x_{i}} x_{n} (x_{1} x_{n}) d x_{1} . . . . . . d x_{n} = \int_{- x}^{\infty} \int_{- \infty}^{x_{0} + a} \frac{n!}{(n - 2)!} {[F (x_{n}) - F (x_{1})]}^{n - 2} f (x_{1}) f (x_{n}) d x_{1} d x_{n}$

Putting $y = F (x_{n}) - F (x_{1}), d y = f (x_{n}) d x_{n}$

We get,

$\int_{- x}^{\infty} \int_{x}^{y + a} {[F (x_{n}) - F (x_{1})]}^{n - 2} f (x_{n}) d x_{n} = \int_{- \infty}^{\infty} \int_{0}^{F (x_{1} + a) - F (x_{1})} y^{n - 2} d y = \int_{- \infty}^{\infty} \frac{1}{n - 1} {[F (x_{1} + a) - F (x_{1})]}^{n - 1}$

Thus, $P (R \leq a) = n \int_{-}^{\infty} {[F (x_{1} + a) - F (x_{1})]}^{- 1} f (x_{1}) d x_{1}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Compute the density of the range of a sample of size $n$ from a continuous distribution having density function $f$ .

Short Answer

Step by step solution

Joint probability distribution :

Explanation :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Calculus

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Compute the density of the range of a sample of size nfrom a continuous distribution having density function f.

Short Answer

Step by step solution

Joint probability distribution :

Explanation :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Calculus

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Compute the density of the range of a sample of size $n$ from a continuous distribution having density function $f$ .