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Chapter 12: Problem 6

Let \(Q=X_{1} X_{2}-X_{3} X_{4}\), where \(X_{1}, X_{2}, X_{3}, X_{4}\) is a random sample of size 4 from a distribution which is \(n\left(0, \sigma^{2}\right) .\) Show that \(Q / \sigma^{2}\) does not have a chi-square distribution. Find the moment-generating function of \(Q / \sigma^{2} .\)

Short Answer

Expert verified
The MGF of \(Q / \sigma^2\) is \(E[e^{t(X_{1}X_{2}-X_{3}X_{4})/\sigma^2}]\) and it is different from the MGF of a Chi-square distribution. Hence, \(Q / \sigma^2\) does not follow a Chi-square distribution.

Step by step solution

01

Define the MGF

The moment-generating function (MGF) is given by \(M(t) = E[e^{tX}]\), where X is a random variable, E is the expectation operator, and \(e^{tX}\) is the exponential function of \(tX\).
02

Calculate the MGF of \(Q / \sigma^2\)

The moment generating function of Q (where Q = \(X_{1}X_{2}-X_{3}X_{4}\)) divided by \(\sigma^2\) is given by \(M_{Q / \sigma^2}(t) = E[e^{t(X_{1}X_{2}-X_{3}X_{4})/\sigma^2}]\). If \(X_{1}, X_{2}, X_{3}, X_{4}\) are independent identically distributed (i.i.d.) normal random variables, then their joint MGF becomes a product of their individual MGFs. Therefore, \(M_{Q / \sigma^2}(t) = M_{X_{1}X_{2}/\sigma^2}(t) \cdot M_{-X_{3}X_{4}/\sigma^2}(t) = E[e^{tX_{1}X_{2}/\sigma^2}] \cdot E[e^{t(-X_{3}X_{4}/\sigma^2)}]\).
03

Compare the MGF with a Chi-square distribution

The MGF of a Chi-square distribution with n degrees of freedom is given by \((1-2t)^{-n/2}\). Compare this with the result from Step 2. We can see that the two MGFs are not equal, hence \(Q / \sigma^2\) does not follow a Chi-square distribution.

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Key Concepts

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

Random Sample
In statistical studies, a random sample is a subset of individuals chosen from a larger set, or population. The idea of randomness means that every individual has an equal chance of being selected from the population. This results in a sample that is representative of the population, minimizing bias in statistical analysis.

For example, consider a population with a normal distribution. If you take a random sample of size four, each element in this sample is drawn independently and has the same probability distribution as the population. This means that each sample point is an independent and identically distributed (i.i.d.) random variable. In the exercise, each of the random variables \(X_1, X_2, X_3,\) and \(X_4\) conforms to a standard normal distribution, having a mean of 0 and a variance of \(\sigma^2\).

To ensure the integrity of statistical analyses like these, it's crucial that the samples are random so that conclusions drawn from them are reliable.
Chi-Square Distribution
The chi-square distribution is commonly used in statistics, especially in hypothesis testing and constructing confidence intervals. It arises when you sum the squares of a set of independent standard normal random variables.

Key aspects of chi-square distribution include:
  • It is non-negative since it results from squaring the values.
  • Its shape is determined by the degrees of freedom \(n\). More degrees of freedom make it more symmetric.
  • The moment-generating function of a chi-square distribution with \(n\) degrees of freedom is \((1-2t)^{-n/2}\).


In the given exercise, the task was to determine if the expression \(Q/\sigma^2\) follows a chi-square distribution by comparing MGFs. The result showed that the MGF of \(Q/\sigma^2\) doesn't match that of a chi-square distribution, verifying it doesn't follow this distribution.
Normal Distribution
The normal distribution, often known as the bell curve due to its shape, is a critical concept in statistics. It describes how the values of a variable are distributed. Values are symmetrically distributed around the mean, making it a model for natural phenomena and measurement errors.
  • Characterized by two parameters: mean \(\mu\) and variance \(\sigma^2\).
  • Approximately 68% of data within one standard deviation (\(\sigma\)) of the mean, 95% within two, and 99.7% within three.
  • The PDF is given by \(\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\).


In the exercise, the random sample \(X_1, X_2, X_3, X_4\) all follow a normal distribution with mean 0 and variance \(\sigma^2\). This forms the basis to explore further statistical properties and determine the moment-generating function of combined expressions, as seen with \(Q\). Understanding normal distribution is vital because it's foundational for other statistical concepts, like the Central Limit Theorem.

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Most popular questions from this chapter

Let \(\mathbf{A}=\left[a_{i j}\right]\) be a real symmetric matrix. Prove that \(\sum_{j} \sum_{i} a_{i j}^{2}\) is equal to the sum of the squares of the characteristic numbers of A. Hint. If \(\mathbf{L}\) is an orthogonal matrix, show that \(\sum_{f} \sum_{1} a_{i j}^{2}=\operatorname{tr}\left(\mathbf{A}^{2}\right)=\operatorname{tr}\left(\mathbf{L}^{\prime} \mathbf{A}^{2} \mathbf{L}\right)=\) \(\operatorname{tr}\left[\left(\mathbf{L}^{\prime} \mathbf{A} \mathbf{L}\right)\left(\mathbf{L}^{\prime} \mathbf{A} \mathbf{L}\right)\right]\)

Let \(\bar{X}\) and \(S^{2}\) denote, respectively, the mean and the variance of a random sample of size \(n\) from a distribution which is \(n\left(0, \sigma^{2}\right\rangle .\) (a) If A denotes the symmetric matrix of \(n \bar{X}^{2}\), show that \(\mathbf{A}=(1 / n) \mathbf{P}\), where \(\mathbf{P}\) is the \(n \times n\) matrix, each of whose elements is equal to one. (b) Demonstrate that \(\mathrm{A}\) is idempotent and that the \(\operatorname{tr} \mathbf{A}=1\). Thus \(n \bar{X}^{2} / \sigma^{2}\) is \(x^{2}(1) .\) (c) Show that the symmetric matrix \(\mathbf{B}\) of \(n S^{2}\) is \(\mathbf{I}-(1 / n) \mathbf{P} .\) (d) Demonstrate that \(\mathbf{B}\) is idempotent and that \(\operatorname{tr} \mathbf{B}=n-1 .\) Thus \(n S^{2} / \sigma^{2}\) is \(\chi^{2}(n-1)\), as previously proved otherwise. (e) Show that the product matrix \(\mathbf{A B}\) is the zero matrix.

Let \(n=2\) and take $$ \mathbf{V}=\left[\begin{array}{cc} \sigma_{1}^{2} & \rho \sigma_{1} \sigma_{2} \\ \rho \sigma_{1} \sigma_{2} & \sigma_{2}^{2} \end{array}\right] $$ Determine \(|\mathbf{V}|, \mathbf{V}^{-1}\), and \((\mathbf{x}-\boldsymbol{\mu})^{\prime} \mathbf{V}^{-1}(\mathbf{x}-\boldsymbol{\mu})\). Compare the bivariate normal p.d.f. with the multivariate normal p.d.f. when \(n=2\).

Let \(X_{1}, X_{2}, \ldots, X_{n}\) have a multivariate normal distribution with positive definite covariance matrix \(\mathbf{V}\). Prove that these random variables are mutually stochastically independent if and only if \(\mathbf{V}\) is a diagonal matrix.

Let \(Q_{1}\) and \(Q_{2}\) be two nonnegative quadratic forms in the items of a random sample from a distribution which is \(n\left(0, \sigma^{2}\right) .\) Show that another quadratic form \(Q\) is stochastically independent of \(Q_{1}+Q_{2}\) if and only if \(Q\) is stochastically independent of each of \(Q_{1}\) and \(Q_{2} .\) Hint. Consider the orthogonal transformation that diagonalizes the matrix of \(Q_{1}+Q_{2}\). After this transformation, what are the forms of the matrices of \(Q, Q_{1}\), and \(Q_{2}\) if \(Q\) and \(Q_{1}+Q_{2}\) are stochastically independent?
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