Problem 8
Question
Let \(V\) and \(U\) be independent chi square random variables with 7 and 9 degrees of freedom, respectively. Is it more likely that \(\frac{V / 7}{U / 9}\) will be between (1) \(2.51\) and \(3.29\) or (2) \(3.29\) and 4.20?
Step-by-Step Solution
Verified Answer
To identify which range is more likely, calculate the differences in CDF values for both ranges. The range with the larger difference in CDF values is the more likely range.
1Step 1: Identify the F-distribution
The F-distribution is defined as the ratio of two normalized independent chi-square random variables. The ratio \(\frac{V / 7}{U / 9}\) follows an F-distribution with 7 and 9 degrees of freedom. The degrees of freedom for the numerator and denominator are derived from the degrees of freedom of \(V\) and \(U\) respectively.
2Step 2: Use the CDF for the F-distribution
The cumulative distribution function (CDF) of the F-distribution gives the probability that a random variable with that distribution will take on a value less than or equal to the input. In this case, to find the probability that the mentioned ratio is between (1) \(2.51\) and \(3.29\), and (2) \(3.29\) and \(4.20\), subtract the CDF value corresponding to the lower bound from that of the upper bound.
3Step 3: Compare the Two Probabilities
Compute the probabilities for both ranges and compare them. The range with the higher probability is more likely.
Key Concepts
Chi-Square Random VariablesDegrees of FreedomCumulative Distribution Function (CDF)
Chi-Square Random Variables
Chi-square random variables play a significant role in statistics, particularly in hypothesis testing and constructing confidence intervals. These variables arise from the sum of the squares of independent standard normal random variables. For example, if you have a variable, say Z, which is normally distributed with a mean of 0 and a variance of 1 (a standard normal distribution), and you square this variable, Z2, it becomes a chi-square distributed variable with 1 degree of freedom.
A key characteristic of a chi-square distribution is that it is always positive and skewed to the right, since squaring eliminates any negative signs. The shape of the chi-square distribution depends on the degrees of freedom; as the degrees of freedom increase, the distribution becomes less skewed and more bell-shaped. This is important in our exercise with variables V and U, which are chi-square distributed with 7 and 9 degrees of freedom, respectively, as the ratio of these variables follows an F-distribution.
A key characteristic of a chi-square distribution is that it is always positive and skewed to the right, since squaring eliminates any negative signs. The shape of the chi-square distribution depends on the degrees of freedom; as the degrees of freedom increase, the distribution becomes less skewed and more bell-shaped. This is important in our exercise with variables V and U, which are chi-square distributed with 7 and 9 degrees of freedom, respectively, as the ratio of these variables follows an F-distribution.
Degrees of Freedom
Degrees of freedom are an essential concept in statistics, often noted in chi-square and F-distributions. Essentially, they represent the number of independent values in a calculation that are free to vary. To put it simply, if you are calculating the mean of 4 numbers, you are free to choose the first three values arbitrarily, but the last value is constrained by the choice of the first three if you want to reach a particular mean. Hence, you have 3 degrees of freedom in this case.
In the context of chi-square variables like in our exercise, the degrees of freedom correspond to the number of standard normal variables being summed. For V, with 7 degrees of freedom, we would be summing the squares of 7 independent standard normal variables. The degrees of freedom help to determine the shape of the chi-square distribution, which, in turn, affects the behavior of the resulting F-distribution. When using these distributions for statistical tests, it's important to use the correct degrees of freedom to obtain accurate p-values or critical values.
In the context of chi-square variables like in our exercise, the degrees of freedom correspond to the number of standard normal variables being summed. For V, with 7 degrees of freedom, we would be summing the squares of 7 independent standard normal variables. The degrees of freedom help to determine the shape of the chi-square distribution, which, in turn, affects the behavior of the resulting F-distribution. When using these distributions for statistical tests, it's important to use the correct degrees of freedom to obtain accurate p-values or critical values.
Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) is a fundamental tool within probability theory and statistics. It provides the probability that a random variable X will take a value less than or equal to x. In mathematical terms, for a random variable X, the CDF, F(x), is defined as F(x) = P(X ≤ x).
A CDF graph rises from 0 to 1, as it encompasses all possible outcomes of the random variable. Visually, it provides a way to see the likelihood of observing certain outcomes or ranges of outcomes. In the given exercise, utilizing the CDF for the F-distribution allows us to compute the probabilities that the F-statistic falls within certain ranges. By comparing the CDF values at different points, specifically the bounds of the given intervals, we can determine the likelihoods of the F-statistic being within those intervals. This is exactly what is required to find which of the two provided intervals for the ratio V/7 over U/9 is more likely to occur.
A CDF graph rises from 0 to 1, as it encompasses all possible outcomes of the random variable. Visually, it provides a way to see the likelihood of observing certain outcomes or ranges of outcomes. In the given exercise, utilizing the CDF for the F-distribution allows us to compute the probabilities that the F-statistic falls within certain ranges. By comparing the CDF values at different points, specifically the bounds of the given intervals, we can determine the likelihoods of the F-statistic being within those intervals. This is exactly what is required to find which of the two provided intervals for the ratio V/7 over U/9 is more likely to occur.
Other exercises in this chapter
Problem 4
Use the fact that \((n-1) S^{2} / \sigma^{2}\) is a chi square random variable with \(n-1\) df to prove that $$ \operatorname{Var}\left(S^{2}\right)=\frac{2 \si
View solution Problem 6
If \(Y\) is a chi square random variable with \(n\) degrees of freedom, the pdf of \((Y-n) / \sqrt{2 n}\) converges to \(f_{Z}(z)\) as \(n\) goes to infinity (r
View solution Problem 11
If the random variable \(F\) has an \(F\) distribution with \(m\) and \(n\) degrees of freedom, show that \(1 / F\) has an \(F\) distribution with \(n\) and \(m
View solution Problem 13
Show that as \(n \rightarrow \infty\), the pdf of a Student \(t\) random variable with \(n\) df converges to \(f_{Z}(z)\). (Hint: To show that the constant term
View solution