Problem 94 - Applied Statistics and Probability for Engineers

Question

The chi-squared random variable with \(k\) degrees of freedom has moment- generating function \(M_{X}(t)=(1-2 t)-k / 2\). Suppose that \(X_{1}\) and \(X_{2}\) are independent chi-squared random variables with \(k_{1}\) and \(k_{2}\) degrees of freedom, respectively. What is the distribution of \(Y=X_{1}+X_{2} ?\)

Step-by-Step Solution

Answer

Y follows a chi-squared distribution with \(k_1 + k_2\) degrees of freedom.

Step 1: Understand the Moment-Generating Function

The moment-generating function (MGF) of a chi-squared random variable with \(k\) degrees of freedom is given by \(M_{X}(t) = (1-2t)^{-k/2}\). This MGF is crucial because it uniquely defines the distribution of the random variable.

Step 2: Find the MGF of the Sum

Since \(X_1\) and \(X_2\) are independent, the MGF of their sum \(Y = X_1 + X_2\) is the product of their MGFs. Thus, \(M_Y(t) = M_{X_1}(t) \times M_{X_2}(t) = (1-2t)^{-k_1/2} \times (1-2t)^{-k_2/2}\).

Step 3: Simplify the MGF

Combine the exponents using the property of exponents. \(M_Y(t) = (1-2t)^{-(k_1 + k_2)/2}\). This is the MGF of a chi-squared distribution with \(k_1 + k_2\) degrees of freedom.

Step 4: Identify the Distribution of Y

The simplified MGF \((1-2t)^{-(k_1+k_2)/2}\) corresponds to the chi-squared distribution with \(k_1 + k_2\) degrees of freedom. Hence, \(Y\) follows a chi-squared distribution with \(k_1 + k_2\) degrees of freedom.

Key Concepts

Moment-Generating Function

The moment-generating function (MGF) is a powerful tool in probability and statistics. It helps to describe the distribution of a random variable. For a chi-squared random variable with degrees of freedom denoted by \( k \), the MGF is expressed as \( M_{X}(t)=(1-2t)^{-k/2} \). This formula provides a compact way of summarizing the characteristics of the chi-squared random variable.

The key purpose of the MGF is that it can uniquely defines a distribution. The MGF is not just for chi-squared variables; it's applicable more broadly, but is particularly neat for distributions like the chi-squared. The property that makes MGFs especially useful is that for a sum of independent random variables, the MGF of the sum is the product of their MGFs. This allows us to handle sums of independent chi-squared variables easily by multiplying their MGFs.

So, when you have \( X_1 \) and \( X_2 \) as independent chi-squared random variables, their joint distribution can be studied using the MGFs. MGF makes analysis cleaner and insightful, especially in solving how sums of random variables behave.

Degrees of Freedom

Degrees of freedom, often abbreviated as \( k \), is a parameter that critically influences the shape of the chi-squared distribution. In statistical terms, degrees of freedom refer to the number of independent values or quantities which can be assigned to a statistical distribution. Basically, it is a matter of how much data you have versus how many parameters you are free to vary.

For a chi-squared distribution, the degrees of freedom determine how "spread out" or "peaked" the distribution is. As \( k \) increases, the chi-squared distribution approaches a more normal shape. Understanding degrees of freedom is important because it helps in fitting the right type of chi-squared distribution to data.

In the context of the original problem, when two chi-squared random variables \( X_1 \) and \( X_2 \) are summed, their degrees of freedom are simply added together, \( k_1 + k_2 \). Thus, the resulting distribution \( Y = X_1 + X_2 \) has \( k_1 + k_2 \) degrees of freedom. This is a straightforward yet powerful property that simplifies the understanding and calculation of these distributions.

Chi-Squared Random Variable

A chi-squared random variable is a special type of random variable used extensively in statistical analyses. It arises by summing the squares of \( k \) independent standard normal random variables. This is particularly useful in hypothesis testing, for instance in tests of independence using contingency tables or in goodness-of-fit tests.

The importance of chi-squared random variables stems from their distribution which quickly becomes asymmetric especially at lower degrees of freedom, and then gradually approaches symmetry with higher \( k \). You might encounter chi-squared variables when working with variance estimates, because they often arise naturally when dealing with the sampling distribution of variance estimates for normally distributed data.

When adding independent chi-squared random variables together, like \( X_1 \) and \( X_2 \) in the problem, the sum \( Y = X_1 + X_2 \) is also a chi-squared random variable. The degrees of freedom of \( Y \) will equal the sum of the individual degrees of freedom, as we see \( Y \) has \( k_1 + k_2 \) degrees of freedom. This property keeps calculations elegant and helps in predicting how the sum behaves statistically.