Question

Let \(X_{1}, X_{2}, X_{3}, X_{4}\) denote a random sample of size 4 from a distribution that is \(N\left(0, \sigma^{2}\right)\). Let \(Y=\sum_{1}^{4} a_{i} X_{i}\), where \(a_{1}, a_{2}, a_{3}\), and \(a_{4}\) are real constants. If \(Y^{2}\) and \(Q=X_{1} X_{2}-X_{3} X_{4}\) are independent, determine \(a_{1}, a_{2}, a_{3}\), and \(a_{4}\).

Short answer

One possible solution can be \(a_1 = a_4 = 1\) and \(a_2 = -1, a_3 = 1\). There may be other sets of solutions as well, as long as they satisfy the equations \(a_1 = a_4\) and \(a_2 = - a_3\).

Step-by-step solution

Calculation of Y

We should first declare \(Y\) and \(Q\) using the given formulas. \(Y = a_1 X_1 + a_2 X_2 + a_3 X_3 + a_4 X_4\) and \(Q = X_1 X_2 - X_3 X_4\).

Property of Independence of Normal Variables

Because \(Y\) and \(Q\) are known to be independent, the covariance between these two variables should be zero. The covariance between two variables are zero if their expected product is the product of their expectations. Therefore, \(E[YQ] = E[Y] E[Q]\).

Calculation of Expected Values

By substituting our declared variables into the equation, we get \(E[a_1 X_1 X_1 X_2 - a_1 X_1 X_3 X_4 + a_2 X_2 X_1 X_2 - a_2 X_2 X_3 X_4 + a_3 X_3 X_1 X_2 - a_3 X_3 X_3 X_4 + a_4 X_4 X_1 X_2 - a_4 X_4 X_3 X_4] = 0\). Note that since \(X_1, X_2, X_3, X_4\) are independent identically distributed normal random variables and the expectation \(E(X_i X_j)\) equals to zero except for when \(i=j\). Simplify the equation to get \(a_1 E[X_1^2 X_2 - X_1 X_3 X_4] + a_2 E[X_2^2 X_1 - X_2 X_3 X_4] + a_3 E[X_3 X_1 X_2 - X_3^2 X_4] + a_4 E[X_4 X_1 X_2 - X_4 X_3 X_4] = 0\). Because, \(E(X_i X_j)\) equals to zero, the only non-zero terms left will be where \(i\) equals to \(j\). This simplifies the equation to \(a_1 \sigma^2 - a_4 \sigma^2 = 0\).

Determination of coefficients

Now, we solve the equation, we get \(a_1 = a_4\). Also, \(Y^2\) and \(Q\) are rotationally symmetric, which indicates that sign can be reversed without affecting the distribution. Thus, \(a_2 = - a_3\). With these constraints, there are infinite solutions to \(a_i\)s as long as they satisfy \(a_1 = a_4\) and \(a_2 = - a_3\). One possible solution can be \(a_1 = a_4 = 1\) and \(a_2 = -1, a_3 = 1\).

Key concepts

Random Sample

A random sample is a fundamental concept in statistical analysis, representing a subset of individual observations from a larger population. The key feature of this sample is that each individual is chosen randomly, ensuring each one's chance of being selected is equally likely. This randomness is imperative because it helps to eliminate biases and provides a representative cross-section of the larger population.

When working with random samples, especially in the context of mathematical statistics, one of the primary goals is to make inferences about the population from which the sample is drawn. It's important to note that each element of the random sample, like the random variables denoted as \( X_{1}, X_{2}, X_{3}, X_{4} \) in our example, should ideally be independent and identically distributed (i.i.d). This condition is crucial when trying to apply certain theorems and methods, like the Central Limit Theorem, which allows for the approximation of the sampling distribution of the sum of these random variables, regardless of the population's distribution, given that the sample size is sufficiently large.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, this distribution will appear as a bell curve.

In the discipline of mathematical statistics, the normal distribution is so widely utilized because of its convenient mathematical properties and the Central Limit Theorem's implication that many complex datasets can be modeled using this distribution, provided the sample size is large enough. Our exercise involves a distribution that is \( N(0, \$sigma^2) \), indicating a normal distribution with a mean of 0 and a variance of \( \$sigma^2 \). This bell-shaped curve is essential in the context of our question as it simplifies many operations such as transformations and computation of probabilities.

Independence of Random Variables

Independence is a cornerstone in probability theory and mathematical statistics. When two random variables are independent, the outcome of one does not influence the outcome of another. For independent random variables, the probability of occurrence of one variable's outcomes does neither increase nor decrease the likelihood of occurrence of the other variable's outcomes.

In relation to our exercise, the property of independence is pivotal when considering the relationship between \( Y^2 \) and \( Q \). Because these two are independent, their covariance must be zero, which implies that the expected value of their product equals the product of their expected values, that is, \( E[YQ] = E[Y]E[Q] \). This property allows for simplification and resolution in determining the constants \( a_{1}, a_{2}, a_{3} \), and \( a_{4} \), which refers back to the initial stipulation that the variables \( X_1, X_2, X_3, \) and \( X_4 \) in the sum \( Y \) are independent - a subtle nuance that might have been overlooked without a proper understanding of the concept of independent random variables.