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Chapter 6: Problem 57

Let \(X\) have an \(F\) distribution with \(v_{1}\) numerator df and \(v_{2}\) denominator df. a. Determine the mean value of \(X\). b. Determine the variance of \(X\).

Short Answer

Expert verified
The mean is \( \frac{v_2}{v_2 - 2} \) if \( v_2 > 2 \), and the variance is \( \frac{2v_2^2(v_1 + v2 - 2)}{v_1(v_2 - 2)^2(v_2 - 4)} \) if \( v_2 > 4 \).

Step by step solution

01

Understanding the F-distribution

The F-distribution is a probability distribution that arises from comparing two variances. It is defined by its two degrees of freedom: the numerator degrees of freedom ( v_1 ) and the denominator degrees of freedom ( v_2 ). The F-distribution is used extensively in analysis of variance and hypothesis testing.
02

Mean of F-distribution

The mean of a random variable with an F-distribution is calculated using the formula:\[ \text{Mean} = \frac{v_2}{v_2 - 2} \]provided that the denominator degrees of freedom ( v_2 ) is greater than 2. Otherwise, the mean is not defined.
03

Variance of F-distribution

The variance of a random variable with an F-distribution is calculated using the formula:\[ \text{Variance} = \frac{2v_2^2(v_1 + v_2 - 2)}{v_1(v_2 - 2)^2(v_2-4)} \]provided that v_2 > 4 . If v_2 is less than or equal to 4, the variance is not defined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degrees of Freedom
Degrees of freedom (df) are the number of independent values that can vary in an analysis without breaking any constraints. In the context of the F-distribution, degrees of freedom are crucial for determining the shape of the distribution. You will encounter two types:
  • Numerator degrees of freedom (\( v_1 \)): These determine variability in the data or model that you are comparing.
  • Denominator degrees of freedom (\( v_2 \)): These are used to estimate the mean of the population you're comparing.
The degrees of freedom impact the critical values in statistical testing, like the F-test. A higher degree of freedom typically means a smoother distribution. Understanding and calculating these accurately is key to making appropriate statistical inferences.
Mean of F-distribution
The mean of the F-distribution gives us information about the central location of the distribution. However, it is only defined when the denominator degrees of freedom (\( v_2 \)) is greater than 2. The formula to compute the mean is:\[\text{Mean} = \frac{v_2}{v_2 - 2}\]Here's why it works:
  • If \( v_2 \) is small (< 3), you can't calculate a mean because the distribution is too skewed.
  • As \( v_2 \) increases, the central limit theorem kicks in, making the mean well-defined.
When you have sufficient degrees of freedom, this ensures the distribution is more balanced and can be summarized by a mean value. Remember this requirement when evaluating statistics using an F-distribution.
Variance of F-distribution
Variance measures how spread out the values are in the F-distribution. For the F-distribution, variance is more complex because it's not always defined. You can only use the formula if the denominator degrees of freedom (\( v_2 \)) exceeds 4:\[\text{Variance} = \frac{2v_2^2(v_1 + v_2 - 2)}{v_1(v_2 - 2)^2(v_2-4)}\]The variance depends heavily on both the numerator (\( v_1 \)) and denominator (\( v_2 \)) degrees of freedom. Here are critical points to keep in mind:
  • When \( v_2 \leq 4 \), variance is undefined, as the distribution becomes too erratic.
  • For higher \( v_2 \), the variance reflects typical distribution measurement 鈥� the larger the \( v_1 \), the more spread out values.
Make sure to verify these conditions when dealing with statistical models involving F-distributions, to avoid incorrect inferences.

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Most popular questions from this chapter

Suppose the distribution of the time \(X\) (in hours) spent by students at a certain university on a particular project is gamma with parameters \(\alpha=50\) and \(\beta=2\). Because \(\alpha\) is large, it can be shown that \(X\) has approximately a normal distribution. Use this fact to compute the probability that a randomly selected student spends at most \(125 \mathrm{~h}\) on the project.

Suppose the sediment density \((\mathrm{g} / \mathrm{cm})\) of a randomly selected specimen from a region is normally distributed with mean \(2.65\) and standard deviation \(.85\) (suggested in "Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants,"Water Res., 1984: 1169-1174). a. If a random sample of 25 specimens is selected, what is the probability that the sample average sediment density is at most \(3.00\) ? Between \(2.65\) and \(3.00\) ? b. How large a sample size would be required to ensure that the first probability in part (a) is at least .99?

The mean weight of luggage checked by a randomly selected tourist-class passenger flying between two cities on a certain airline is \(40 \mathrm{lb}\), and the standard deviation is \(10 \mathrm{lb}\). The mean and standard deviation for a business-class passenger are \(30 \mathrm{lb}\) and \(6 \mathrm{lb}\), respectively. a. If there are 12 business-class passengers and 50 tourist-class passengers on a particular flight, what are the expected value of total luggage weight and the standard deviation of total luggage weight? b. If individual luggage weights are independent, normally distributed rv's, what is the probability that total luggage weight is at most \(2500 \mathrm{lb}\) ?

The difference of two independent normal variables itself has a nomal distribution. Is it true that the difference between two independent chi- squared variables has a chi-squared distribution? Explain.

Two airplanes are flying in the same direction in adjacent parallel corridors. At time \(t=0\), the first airplane is \(10 \mathrm{~km}\) ahead of the second one. Suppose the speed of the first plane \((\mathrm{km} / \mathrm{h})\) is normally distributed with mean 520 and standard deviation 10 and the second plane's speed, independent of the first, is also normally distributed with mean and standard deviation 500 and 10 , respectively. a. What is the probability that after \(2 \mathrm{~h}\) of flying, the second plane has not caught up to the first plane? b. Determine the probability that the planes are separated by at most \(10 \mathrm{~km}\) after \(2 \mathrm{~h}\).
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