Question

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

The values of \(b\) and \(c\) for a uniform distribution that has the same mean and same variance as a chi-square distribution with 8 degrees of freedom are \(b = 8 - 2\sqrt{3}\) and \(c = 8 + 2\sqrt{3}\) respectively.

Step-by-step solution

Define and Equate the Means

The mean of the Chi-square distribution is \(8\). The mean of a uniform distribution is \((b+c)/2\). Equating these values will give us the equation \((b+c)/2 = 8\).

Define and Equate the Variances

The variance of the Chi-square distribution is \(16\). The variance of a uniform distribution is \((c-b)^2/12\). By equating these values, we get the second equation \((c-b)^2/12 = 16\).

Solve for 'b' and 'c'

We now have a system of two equations that needs to be solved for ‘b’ and ‘c’. The equation from step 2 can be simplified to \(c-b = 4\sqrt{3}\). By adding this value to ‘b’ in the equation from step 1, we find 'c'. Similarly, by subtracting this value from ‘c’ in the equation from step 1, we find 'b'.

Key concepts

Continuous Probability Distribution

Understanding a continuous probability distribution is crucial when dealing with variables that can take an infinite number of values within a given range. Unlike discrete distributions that are concerned with variables that have specific, countable outcomes, a continuous distribution is about variables that can be any value within certain limits.

A classic representation of a continuous distribution is the uniform distribution. Think of the uniform distribution as an evenly stretched string across an interval of numbers where every point is equally likely to occur. The density of this 'string' is constant, meaning there's no bias towards any segment of the interval. In the case of the textbook exercise, the interval is given by two points, 'b' and 'c'.

When you hear 'uniform distribution of the continuous type,' picture a flat line from point 'b' to 'c' on a graph. The height of the line represents the probability density, and since it's a flat line, it means the height is the same anywhere along the line. The mean (average value) of this distribution is exactly at the midpoint of 'b' and 'c', and variance (measure of dispersion) is related to how spread out 'b' and 'c' are from this mean.

Chi-square Distribution

The Chi-square distribution is quite a versatile character in statistics, especially when it comes to assessing the goodness-of-fit and conducting hypothesis testing for variance. This distribution exclusively appears in positive values and has just one parameter controlling its shape, which is the 'degrees of freedom'.

What's unique about the Chi-square distribution is that its mean and variance are twice the value of its degrees of freedom. This gives us a handy relationship when we want to match its mean and variance to those of a uniform distribution. In the textbook problem, the Chi-square distribution with 8 degrees of freedom has a mean of 8 and a variance of 16. These values become the target for finding the equivalent 'b' and 'c' values for the uniform distribution.

The Chi-square distribution is skewed to the right and its shape becomes more symmetrical as the degrees of freedom increase. It's not a flat like the uniform distribution but has a peak that starts at zero and goes down as the value on the X-axis increases.

Degrees of Freedom

Degrees of freedom, often abbreviated as df, are an essential concept in statistics, providing a measure of the number of independent values or quantities that can vary in an analysis without breaking any constraints. In the Chi-square distribution, the degrees of freedom essentially shape the distribution, influencing its mean and variance.

The more degrees of freedom you have, the closer the Chi-square distribution gets to a normal distribution, although it still remains positively skewed. This mainly comes up in the context of various statistical tests, where degrees of freedom are used to determine critical values from the Chi-square distribution. In our exercise, 8 degrees of freedom determine both the shape of the Chi-square distribution and, interestingly, give us a starting point to find our uniform distribution's 'b' and 'c'.

It's important to recognize that degrees of freedom are also associated with the concept of estimators in statistical models. It determines the number of values in the final calculation of a statistic that are free to vary. Estimators with more degrees of freedom are generally preferred as they tend to be more stable and provide more accurate model fits.