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 \(\beta>0\) 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 independence of \(Z_{1}=Y_{1}\) and \(Z_{2}=Y_{2}-Y_{1}\) characterizes the gamma pdf \(f(x)\), but with \(\alpha=1\) and \(\beta=1\), not \(\beta>0\), as stated in the problem.

Step-by-step solution

Change of variables

Using the change-of-variable technique to find the joint pdf of \(Z_{1}\) and \(Z_{2}\), we start by setting up the Jacobian matrix of the transformation from \(Y_{1}, Y_{2}\) to \(Z_{1}, Z_{2}\). The equations are \(Z_{1} = Y_{1}\) and \(Z_{2} = Y_{2} - Y_{1}\). Therefore, the Jacobian is \(J = 1\). Now we obtain the joint pdf \(g(z_{1},z_{2})\) as \(g(z_{1},z_{2}) = f_{Z_{1},Z_{2}}(z_{1},z_{2}) = f_{Y_{1},Y_{2}}(y_{1},y_{2})|J|\).

Applying the transformation

Apply these transformations to \(f_{Y_{1},Y_{2}}(y_{1},y_{2})\). As \(Y_{1}, Y_{2}\) are order statistics from a two-point sample, it's known that their joint pdf is \(f_{Y_{1},Y_{2}}(y_{1},y_{2}) = 2f(y_{1})f(y_{2}-y_{1})\) for \(0<y_{1}<y_{2}\). Sub in our transformations we get \(g(z_{1}, z_{2}) = 2f(z_{1})f(z_{2})\).

Find expression for g

Now we separate \(g(z_{1},z_{2})\) into \(g_{1}(z_{1})g_{2}(z_{2})\). This can be done because \(Z_{1}\) and \(Z_{2}\) are independent, so their joint pdf can be expressed as the product of their marginal pdfs. We know the functional form of \(g_{1}(z_{1})\), since it is a marginal pdf of an order statistic, therefore, \(g_{1}(z_{1})=2z_{1}e^{-z_{1}}\) for \(z_{1}>0\).

Compute g2(z2)

By comparing \(g_{1}(z_{1})g_{2}(z_{2})\) with \(g(z_{1},z_{2}) = 2f(z_{1})f(z_{2})\), we can find \(g_{2}(z_{2})\) using the solution of the functional equation given. This gives \(g_{2}(z_{2}) = e^{-z_{2}}\) for \(z_{2}>0\).

Draw the conclusion

Given that \(g_{1}(z_{1}) = 2z_{1}e^{-z_{1}}\) and \(g_{2}(z_{2}) = e^{-z_{2}}\), it's clear that \(g_{1}(z_{1})\) is a gamma pdf with \(\alpha=2\) and \(\beta =1\), while \(g_{2}(z_{2})\) is a gamma pdf with \(\alpha=1\) and \(\beta =1\). Hence independence of \(Z_{1}=Y_{1}\) and \(Z_{2}=Y_{2}-Y_{1}\) characterizes the gamma pdf \(f(x)\), but with \(\alpha=1\) and \(\beta=1\), not \(\beta>0\) as stated in the problem.

Key concepts

Gamma Probability Density Function

The gamma probability density function (pdf) is a versatile tool in statistical analysis, particularly useful for modeling the waiting times between events in a Poisson process. Its flexibility comes from two parameters, \( \alpha \) and \( \beta \)—where \( \alpha \) is the shape parameter, and \( \beta \) is the rate (or alternatively, the scale) parameter. The gamma pdf is defined only for \( x > 0 \) and is expressed mathematically as:

\[\begin{equation}f(x;\alpha,\beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}\end{equation}\]
where \( \Gamma(\alpha) \) is the gamma function, serving as a generalization of the factorial function to continuous values.

In the context of order statistics, the gamma pdf is particularly notable because it can express the distribution of the \( k \)th smallest value in a sample. Moreover, certain transformations preserve the gamma nature of a distribution, allowing us to manipulate random variables while maintaining their distributional properties. This characteristic was highlighted in the exercise, with the transformation resulting in a gamma distribution with parameters \( \alpha=1 \) and \( \beta=1 \) under the premise of independence between the variables \( Z_{1} \) and \( Z_{2} \)—although the exercise found a discrepancy in the latter parameter.

Change-of-Variable Technique

A quintessential tool in probability and statistics is the change-of-variable technique, which allows us to transform a given pdf using a set of functions that relate the new variable to the original one. This is essential when we want to work with more convenient or informative variable forms. In practice, we solve for the Jacobian—a determinant that represents the volume distortion during the transformation—to adjust the original pdf accordingly.

The process involves substituting the original variables with the new ones and multiplying by the absolute value of the Jacobian. For an intuitive grasp, consider a two-dimensional example: when transforming variables \( (x, y) \) to \( (u, v) \) using functions \( u = u(x, y) \) and \( v = v(x, y) \)—the Jacobian \( J \) would be expressed as:

\[\begin{equation}J = det\left(\frac{\partial(u, v)}{\partial(x, y)}\right) = \frac{\partial u}{\partial x}\frac{\partial v}{\partial y} - \frac{\partial u}{\partial y}\frac{\partial v}{\partial x}\end{equation}\]
In the exercise, this technique transformed the joint pdf of \( Y_{1} \)and \( Y_{2} \) into that of \( Z_{1} \) and \( Z_{2} \) seamlessly since the Jacobian was equal to 1, making the process a direct substitution without additional scaling.

Joint Probability Density Function

The joint probability density function (pdf) combines multiple random variables into a single function, reflecting the likelihood of different combinations of their outcomes. In a multi-variable context, the joint pdf is a crucial concept because it captures the relationship between variables. The independence of two random variables is characterized by the joint pdf being the product of their individual pdfs—an important property leveraged in the textbook exercise.

Mathematically, for two independent random variables \( X \) and \( Y \) with pdfs \( f_X(x) \) and \( f_Y(y) \) respectively, their joint pdf is \( f_{X,Y}(x,y) = f_X(x)f_Y(y) \) for all \( x \) and \( y \) within their domains. The transformation outlined in the change-of-variable section led us to a joint pdf for \( Z_{1} \) and \( Z_{2} \) that upheld this separability, reinforcing their independence and aligning with the gamma pdf characterizations for each variable individually.

In practice, finding the joint pdf of order statistics, such as \( Y_{1} \)and \( Y_{2} \) in the exercise, often involves recognizing the dependence structure between them and applying the right techniques such as the change-of-variable technique to simplify and analyze their distribution properties. Such understanding is vital for working on more complex statistical analyses involving dependency and transformations.