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Chapter 3: Problem 17

Find the uniform distribution of the continuous type on the interval \((b, c)\) that has the same mean and the same variance as those of a chi-square distribution with 8 degrees of freedom. That is, find \(b\) and \(c\).

Short Answer

Expert verified
The required interval that gives the uniform distribution similar to the given Chi-square distribution is \((b, c) = (7 - \sqrt{3}, 7 + \sqrt{3})\).

Step by step solution

01

Determine the Mean and Variance of the Chi-Square Distribution

For a Chi-square distribution with 'n' degrees of freedom, the mean is 'n' and the variance is 2*n. Given that the Chi-square distribution has 8 degrees of freedom, its mean is 8 and the variance is 16.
02

Set up Equations for The Mean and Variance of the Uniform Distribution

For a Uniform distribution, the mean and variance formulas are \(\frac{b+c}{2}\) and \(\frac{(c-b)^2}{12}\) respectively. So we set up two equations based on these formulas: \(\frac{b+c}{2} = 8\) and \(\frac{(c-b)^2}{12} = 16\).
03

Solve for the Values of 'b' and 'c'

Solving the two equations yielded from step 2 gives us the values of b and c. From the first equation, we can express 'c' as \(c = 16 - b\). Substituting 'c' into the second equation leads us to the quadratic equation \(b^2 -14b+64=0\). The solutions of this quadratic equation are the values of 'b' and 'c' as these are the only two parameters in a uniform distribution.
04

Find the Solutions for 'b' and 'c'

The solutions of the quadratic equation are \(b = 7 - \sqrt{3}\) and \(c = 7 + \sqrt{3}\). Therefore, the interval \((b, c)\) that gives the uniform distribution similar to the given Chi-square distribution is \((7 - \sqrt{3}, 7 + \sqrt{3})\)

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Most popular questions from this chapter

Let \(U\) and \(V\) be independent random variables, each having a standard normal distribution. Show that the mgf \(E\left(e^{t(U V)}\right)\) of the random variable \(U V\) is \(\left(1-t^{2}\right)^{-1 / 2},-1

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Let the mutually independent random variables \(X_{1}, X_{2}\), and \(X_{3}\) be \(N(0,1)\), \(N(2,4)\), and \(N(-1,1)\), respectively. Compute the probability that exactly two of these three variables are less than zero.
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