Question

Let \(X_{1}\) and \(X_{2}\) be independent random variables. Let \(X_{1}\) and \(Y=X_{1}+X_{2}\) have chi-square distributions with \(r_{1}\) and \(r\) degrees of freedom, respectively. Here \(r_{1}

Short answer

Using the provided hint and the properties of the moment generating function for chi-square distributions, we were able to show that the random variable \(X_{2}\) follows a chi-square distribution with \(r-r_{1}\) degrees of freedom.

Step-by-step solution

Express the Moment Generating Function

By the definition of the moment generating function, we write it as: \(M(t) = E\left(e^{t\left(X_{1}+X_{2}\right)}\right)\). From the independence of \(X_{1}\) and \(X_{2}\), this can be factorized as: \(M(t) = E\left[e^{tX_{1}}\right]E\left[e^{tX_{2}}\right]\).

Use known Moment Generating Functions

For a chi-square random variable with \(k\) degrees of freedom, its moment generating function is given by \(M(t) = (1-2t)^{-k/2}\) whenever \(t < 1/2\). Thus, using the given that both \(X_{1}\) and \(Y\) are chi-square distributed with \(r_{1}\) and \(r\) degrees of freedom respectively, we have: \(E\left[e^{tX_{1}}\right] = (1-2t)^{-r_{1}/2}\) and \(E\left[e^{t\left(X_{1}+X_{2}\right)}\right] = (1-2t)^{-r/2}\). Replace these into the factorized moment generating function in step 1 to get \((1-2t)^{-r_{1}/2} E\left[e^{tX_{2}}\right] = (1-2t)^{-r/2}\).

Calculate the Moment Generating Function of \(X_{2}\)

Rearrange the equation from step 2 to derive the moment generating function for \(X_{2}\): \(E\left[e^{tX_{2}}\right] = (1-2t)^{(r_{1}-r)/2}\).

Identify the distribution of \(X_{2}\)

Recall that the moment generating function of a chi-square distribution with \(k\) degrees of freedom is \((1-2t)^{-k/2}\). Looking at the equation derived in step 3, we recognise this form if we take \(k=r-r_{1}\). Therefore, \(X_{2}\) follows a chi-square distribution with \(r-r_{1}\) degrees of freedom.

Key concepts

Moment Generating Function

The moment generating function (MGF) is a critical statistical tool that encapsulates all moments (i.e., expected values of powers) of a random variable. It is defined as \( M(t) = E(e^{tX}) \) , where \( X \) is the random variable and \( t \) is a real number such that the expectation exists.

The MGF is particularly useful because it provides a succinct mathematical expression from which we can derive many properties of the distribution. For instance, by taking derivatives of the MGF with respect to \( t \) and evaluating at zero, one can obtain all the moments of the random variable, such as mean and variance. Moreover, if two random variables have the same MGF (and the MGF exists for both), it implies that they have the same distribution.

In the context of chi-square distributions, the MGF takes on a specific form, \( (1 - 2t)^{-k/2} \) , where \( k \) is the degrees of freedom of the chi-square distribution. The MGF becomes invaluable when dealing with sums of independent random variables, as it helps to identify the distribution resulting from the sum.

Degrees of Freedom

The concept of degrees of freedom (df) in statistics often refers to the number of values in a calculation that are free to vary without violating any imposed constraints. It is an essential aspect of various distributions, among which the chi-square distribution is a notable example.

Ideally, the df relates to the number of independent observations in a dataset minus the number of parameters estimated. In the chi-square distribution, df denotes the number of independent variables with standard normal distributions squared and summed to form the chi-square statistic. Notably, the df characterizes the shape and spread of the chi-square distribution, influencing its skewness and kurtosis.

In the exercise provided, the difference in degrees of freedom \( r - r_1 \) determines the distribution of \( X_2 \) . Knowing that \( X_1 \) has \( r_1 \) degrees of freedom and \( Y \) has \( r \) , we use the subtraction of these df to conclude the resultant distribution of \( X_2 \) . It demonstrates the pivotal role of df in the manipulation and understanding of chi-square distributed variables.

Independent Random Variables

Independent random variables are foundational in probability and statistics, where each variable operates without influence from the others. For random variables \( X \) and \( Y \) , independence means that the occurrence of an event related to \( X \) does not impact the probability of an event related to \( Y \) , and vice versa.

In practical terms, independence allows for the simplification of joint distributions, facilitating calculations, such as those for expected values. If \( X \) and \( Y \) are independent, then the expectation of their product is the product of their expectations: \( E(XY) = E(X)E(Y) \) . This property significantly simplifies the determination of the moment generating function of sums of random variables.

The independence assumption was crucial in the provided exercise, where it allowed the separation of the MGF of the sum \( X_1 + X_2 \) into the product of the MGFs. As a result, the problem became solvable by relating the known MGF of a chi-square distribution with \( r \) degrees of freedom to the MGFs of the contributing independent chi-square random variables \( X_1 \) and \( X_2 \) .