Question

Let \(Y_{1}0\), provided that \(x \geq 0\), and \(f(x)=0\) elsewhere. Show that the independence of \(Z_{1}=Y_{1}\) and \(Z_{2}=Y_{2}-Y_{1}\) characterizes the gamma pdf \(f(x)\), which has parameters \(\alpha=1\) and \(\beta>0 .\) That is, show that \(Y_{1}\) and \(Y_{2}\) are independent if and only if \(f(x)\) is the pdf of a \(\Gamma(1, \beta)\) distribution. Hint: Use the change-of-variable technique to find the joint pdf of \(Z_{1}\) and \(Z_{2}\) from that of \(Y_{1}\) and \(Y_{2}\). Accept the fact that the functional equation \(h(0) h(x+y) \equiv\) \(h(x) h(y)\) has the solution \(h(x)=c_{1} e^{c_{2} x}\), where \(c_{1}\) and \(c_{2}\) are constants.

Short answer

The random variables are independent if they follow a gamma distribution with \(\alpha = 1\) and \(\beta > 0\). To confirm, we transform the joint pdf, compare it to the product of marginal pdfs and ensure they are identical.

Step-by-step solution

Introduce Variables and Acceptance

Given is the random variables \(Z_1 = Y_1\) and \(Z_2 = Y_2 - Y_1\). We accept the fact that \(h(0)h(x+y) = h(x) h(y)\) has a solution in the form \(h(x) = c_1 e^{c_2 x}\), where \(c_1\) and \(c_2\) are constants.

Define the Joint PDF of Z_1 and Z_2

The joint pdf of \(Z_1\) and \(Z_2\), call it \(g(z_1, z_2)\), can be found by applying the change of variables technique. This process involves computing the jacobian determinant and replacing the variables appropriately. The Jacobian matrix is defined as: \( J = \begin{bmatrix} \frac{\partial y_1}{\partial z_1} & \frac{\partial y_1}{\partial z_2} \ \frac{\partial y_2}{\partial z_1} & \frac{\partial y_2}{\partial z_2} \end{bmatrix}\) and \(J = \begin{bmatrix} 1 & 0 \ -1 & 1 \ \end{bmatrix}\). Hence, the Jacobian determinant is equal to \(|J| = 1\).

Transform Joint PDF

Now, the joint pdf of \(Z_1\) and \(Z_2\) can be found by substituting \(Y_1 = Z_1\) and \(Y_2 = Z_1 + Z_2\) into the joint pdf of \(Y_1\) and \(Y_2\) and multiplying by the absolute value of the Jacobian determinant. Remember that \(Y_1\) and \(Y_2\) have a joint pdf by order statistics given by \(f(y_1)f(y_2-y_1)\). We then arrive at the function \(g(z_1, z_2)\).

Apply Independence Criteria

To have independence, the joint pdf of \(Z_1\) and \(Z_2\) should be the product of marginal pdfs. Therefore, we compare \(g(z_1, z_2)\) to \(g(z_1)g(z_2)\). If both are identical, then we see independence.

Conclusion

If the forms are identical, we conclude that the given random variables are independent if and only if \(f(x)\) is the pdf of a gamma distribution with \(\alpha = 1\) and \(\beta > 0\).

Key concepts

Order Statistics

Order statistics is an area within statistics that involves the properties and distributions of the ordered values of a sample. These ordered values are often denoted as Y₁, Y₂, and so on, where Y₁ is the smallest value (minimum) and Y₂ is the second smallest, and they progress in a non-decreasing order. When dealing with order statistics, the joint probability density function of these ordered values holds crucial information regarding their distribution and dependence.

Understanding order statistics is particularly important when dealing with exercises related to the independence of certain combinations of such ordered values. In cases where transformations of these variables are involved, the joint pdf can aid in determining the new relationship between the transformed variables.

Probability Density Function (PDF)

A probability density function (pdf) for a continuous random variable is an equation used to compute probabilities associated with values of the variable. Mathematically expressed, if f(x) is a pdf, then f(x) > 0 for all values in the range of X, and the integral of f(x) over the entire space of X is equal to 1. This implies that the area under the curve of the pdf is a measure of probability.

For the exercise at hand, we focus on the pdf of a gamma distribution characterized by parameters \(\alpha = 1\) and \(\beta > 0\). The gamma distribution is commonly used in reliability analysis and queuing models and has a flexibility that makes it suitable for modeling a wide range of phenomena.

Change-Of-Variable Technique

The change-of-variable technique in probability is used to find the probability distribution of a function of random variables. Essentially, this involves using a set of original variables and transforming them into new variables, generally for simplification or to fit a particular distributional model.

In the given exercise, the original variables are transformed with the definitions \(Z_1 = Y_1\) and \(Z_2 = Y_2 - Y_1\). The technique also requires calculating the Jacobian determinant when multiple variables are involved, which is a scalar value that gives us information about the transformation's scale changes. This determinant is then used to adjust the original joint pdf resulting in the joint pdf of the new variables.

Joint PDF

The joint probability density function (joint pdf) of multiple random variables gives us the probability that each of the variables simultaneously falls within any particular range or discrete set of values specified for that variable. In other words, it offers a way to describe the probability distribution over multiple dimensions or variables simultaneously.

In reference to our exercise, to establish independence between \(Z_1\) and \(Z_2\), one needs to show that the joint pdf of these variables factors into the product of their marginal pdfs. Independence of random variables means that knowing the value of one of the variables gives no information about the value of the other; hence their joint pdf is simply the product of their separate pdfs. Finding the joint pdf is crucial for illustrating whether or not such a relationship of independence holds between the two transformed variables within the context of the gamma distribution.