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Chapter 5: Problem 11

Find the probability that the range of a random sample of size 4 from the uniform distribution having the pdf \(f(x)=1,0

Short Answer

Expert verified
The probability that the range of a random sample of size 4 from the given uniform distribution is less than 1/2, is \( 1/35 \).

Step by step solution

01

Understanding the Problem

In order to determine the probability of the range of a random sample a size of 4 from the uniform distribution is less than 1/2, one must understand that the range is the difference between the largest and smallest number in the random sample set. Therefore, the sample must contain 4 numbers, all of which fall within the range of 0<x<1, so that maximum difference between any two numbers (i.e. the range) is less than 1/2.
02

Setting up the Integral

To find the probability, an integral must be set up. Because of the uniform probability function given, the limits of the integral will be 0 to 1/2. Integrate over the range (0 to 1/2) of \( f(x)=1 \).
03

Calculating the Probability

Calculate the integral of 1 dx from the interval 0 to 1/2. The results given will be 1/2.
04

Adjusting for the Size of the Sample

Because the sample size is 4, the result must be raised to the power of the size of the set. This results in (1/2)^4 = 1/16.
05

Final Calculation

The final calculation will be \(16 \cdot \int_{0}^{1/2} x^3(1-x)^3 dx \). Using standard calculus methods, solve the integral to get \( 1/560 \). Multiply with constant (16), we get \( 16/560 = 2/70 = 1/35 \).

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