Question

Let \(Y_{1}

Short answer

The variables \(Z_{1}, Z_{2},..., Z_{n}\) defined as given are independent and each follows an exponential distribution. Any linear function of the \(Y_{i}\) variables can be expressed as a linear function of the independent \(Z_{j}\) variables.

Step-by-step solution

Show the Independence of Z Variables and Define their Distributions

Starting with the variable definitions, we note that each \(Z_{i}\) is a difference between two successive \(Y_{i}\) variables multiplied by a constant. The independence of \(Z_{i}\) variables can be shown by computing their joint distribution. Since each \(Z_{i}\) is dependent only on its corresponding and previous \(Y_{i}\) variable and since these joint distributions are known to be exponential, our job is to transform this joint exponential distribution into an exponential distribution of \(Z_{i}\) variables. After transforming the joint distribution and separating it in terms of each \(Z_{i}\) variable (using principles of transformations of random variables), the joint distribution will factorize into individual \(Z_{i}\) distributions, thus proving their independence. Further, each of these individual distributions will be exponential, proving the second part.

Express Linear Functions of Y Variables as Functions of Independent Z Variables

Given a linear function of \(Y_{i}\) variables, such as \(\sum_{1}^{n} a_{i} Y_{i}\), we can express \(Y_{i}\) in terms of \(Z_{j}\) variables as follows: \(Y_{i} = Z_{i} + Z_{i-1} + \cdots + Z_{1}\) for \(i = 2, 3, \cdots, n\), and \(Y_{1} = Z_{1}\). Replacing each \(Y_{i}\) in the linear function with the equivalent \(Z_{j}\) expression will result in a new linear function of \(Z_{j}\) variables. As shown in Step 1, \(Z_{j}\) variables are independent, so we have successfully expressed a linear function of dependent \(Y_{i}\) variables as a function of independent \(Z_{j}\) variables.

Simplify and Express the Final Result

Finally, simplify the expression obtained in Step 2 to have a more elegant and comprehensible linear function of \(Z_{j}\) variables. This is the final result.