Question

Let \(Y_{1}

Short answer

a) Variables \(Z_{1}, Z_{2}, ..., Z_{n}\) are defined as certain transformations of order statistics \(Y_{1}, Y_{2}, ..., Y_{n}\) from the exponential distribution. By applying the 'memoryless property' of exponential distribution and Jacobian transformation for transformation of random variables, it can be shown that each \(Z_{i}\) are independent and each follows exponential distribution. b) As \(Y_{i}\) can be re-written in terms of \(Z_{i}\), it implies that any linear functions of \(Y_{i}\) can also be represented as a function of independent variables.

Step-by-step solution

Define the Variables

Firstly, let's define the variables \(Y_{1}<Y_{2}<\cdots<Y_{n}\) as the order statistics of a random sample of size \(n\) from the exponential distribution with pdf \(f(x)=e^{-x}\). Then, we define new variables \(Z_{1}=n Y_{1}, Z_{2}=(n-1)\left(Y_{2}-Y_{1}\right), Z_{3}=(n-2)\left(Y_{3}-Y_{2}\right), \ldots, Z_{n}= Y_{n}-Y_{n-1}\).

Validate the Independence of \(Z_{i}\)

We know exponential distribution has the 'memoryless property'. Due to this property, \(P(Y_{2} > y + Y_{1} | Y_{1} = y) = P(Y_{2} > y)\), for which pdf is \(e^{-y}\). Extending the logic to \(Z_{i}\), we realize \(Z_{i}\) are independent.

Validate the Exponential Distribution of \(Z_{i}\)

The pdf of specified \(Z_{i}\) can be obtained using pdf of \(Y_{i}\), \(Y_{i - 1}\) and Jacobian determinant for transformation from \(Y\) variables to \(Z\) variables. For \(i = 1\), the pdf of \(Z_{1}\) comes out to be \(e^{-y}\) and for \(i > 1\), the pdf of \(Z_{i}\) is \((n - i + 1) e^{-y(n - i + 1)}\) which is the exponential distribution.

Independence of Linear Functions of \(Y_{i}\)

Since \(Y_{i}\) can be written as linear function of \(Z_{i}\) and \(Z_{i}\) are independent, it follows that any linear functions of \(Y_{i}\) can be written as function of independent variables.

Key concepts

Order Statistics

Order statistics play a vital role in a range of statistical analyses and can be crucial in understanding samples from a population. Fundamental to statistics, order statistics are essentially the sorted values from a sample. For instance, given a collection of times taken by different runners to complete a race, ranking them in increasing order would yield the order statistics of that sample.

When analyzing samples, it becomes essential to study the behavior of the extremes - the smallest and largest observations, which are known as the first and last order statistics, respectively. In the context of the exercise, the variables \(Y_{1}\), \(Y_{2}\), ..., \(Y_{n}\) represent the order statistics of a random sample of size \(n\) from an exponential distribution. The transformation into the \(Z_{i}\) variables illuminates the properties of the exponential distribution, paving the way to understanding how these variables interact and how they can be utilized in further statistical applications.

Exponential Distribution

The exponential distribution is frequently encountered in the context of the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. Characterized by its probability density function (pdf) \(f(x)=e^{-x}\) for \(x > 0\), it's a model commonly applied to measure lifetimes of objects or time until an event, like a radioactive particle's decay or the time until a customer's arrival.

The exponential distribution has a unique 'memoryless' property, meaning that the probability of an event occurring in the future is independent of any past events. This property is central to demonstrating the independence of the transformed variables \(Z_{i}\) in our exercise, as it allows us to treat each successive time period as though it were the first. In practical terms, this might mean that the probability a light bulb will burn out is the same regardless of how long it has already been burning.

Probability Density Function (PDF)

The probability density function (PDF) is a fundamental concept in statistics used to specify the probability of a random variable falling within a particular range of values. The PDF essentially describes the distribution of probabilities across the possible values of a continuous random variable. It is the function that, when integrated over an interval, gives the probability that a random variable falls within that interval.

In simpler terms, the PDF is like a recipe that tells us how likely it is to find a random variable at a certain level. For the exponential distribution, the PDF \(f(x)=e^{-x}\) conveys how likely we are to observe a particular time gap between events. In the case of the exercise, the PDF is manipulated through the transformation of variables from \(Y_{i}\) to \(Z_{i}\) variables, which maintains the exponential nature of the distribution for each \(Z_{i}\).

Independent Random Variables

Independence is a cornerstone concept in the study of probability and statistics. Independent random variables are those whose occurrence is not influenced by the occurrence of another. That is, knowing the outcome of one does not provide any information about the outcome of another. For example, rolling a dice and flipping a coin are independent events because the result of the dice roll does not affect the result of the coin flip.

Understanding the independence of random variables is crucial for analyzing complex phenomena, as it allows the simplification of joint probabilities and the separate treatment of individual variables. Within our exercise, we explored the independence of the \(Z_{i}\) variables, which are transformations from the original order statistics of an exponential distribution. This independence is fundamental to our ability to treat complex expressions involving \(Y_{i}\) variables, such as \(\sum_{1}^{n} a_{i} Y_{i}\), as linear functions of independent variables - a profound concept that simplifies analysis and computation in statistics.