Question

Let \(z^{*}\) be drawn at random from the discrete distribution which has mass \(n^{-1}\) at each point \(z_{i}=x_{i}-\bar{x}+\mu_{0}\), where \(\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) is the realization of a random sanple. Determine \(E\left(z^{*}\right)\) and \(V\left(z^{*}\right)\).

Short answer

The expected value of \(z^{*}\) is \(\mu_{0}\) and the variance of \(z^{*}\) is \(\sigma^{2}\).

Step-by-step solution

Define \(z^{*}\) and calculate its expected value

Starting with each point \(z_{i}=x_{i}-\bar{x}+\mu_{0}\), let's calculate the expected value \(E(z^{*})\). Since all \(x_{i}\) have uniform probability of \(1/n\), the expected value becomes \(E\left(z^{*}\right)=\frac{1}{n}\sum_{i=1}^{n}z_{i}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x}+\mu_{0})\), which simplifies to \(\mu_{0}\) with the properties of mean and the definitions of \(x_i\) and \(\bar{x}\).

Calculate the variance of \(z^{*}\)

Now, let's calculate the variance \(V(z^{*})\). Following the definition of variance, we should compute \(V\left(z^{*}\right)=E\left[{(z^{*}-E(z^{*}))}^{2}\right]\). Substituting \(z^{*}\) and \(E(z^{*})\) into the formula, we get \(V\left(z^{*}\right)=E\left[{(x_i-\bar{x})}^{2}\right]=\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^{2}\), which is the variance of the \(x_i\)'s, denoted as \(\sigma^{2}\).

Key concepts

Expected Value

The expected value is a fundamental concept in probability, often denoted as E, which provides a measure of the center or the average outcome one would anticipate from a probability distribution. Imagine you're playing a game that has various outcomes, each with a specific payoff. The expected value will tell you, on average, how much you might win or lose per play in the long run.

To calculate the expected value of a discrete random variable, such as the one represented by z* in our exercise, we sum up the products of each possible value the variable can take and its probability of occurring. In mathematical terms, it is expressed as E(z*) = ∑(value × probability). For our specific problem, since z* have equal chances of being any value z_i, the expected value turns out to be the given constant μ₀.

Variance

Variance measures the spread of a set of numbers, sometimes referred to as their

Random Sample

You may have heard the term random sample tossed around in statistics, but what does it really mean? It's quite simple: a random sample is a subset of individuals chosen from a larger set, or population, where each individual has an equal chance of being selected. Consider it a mini-representation of the entire group.

In our exercise, the set (x₁, x₂, ..., x_n) represents a random sample from a larger population. The random sample serves a crucial role in inferential statistics, where estimates or predictions about the larger population are made based on the data collected from the sample.

However, it is vital that the random sample is just that—random. This ensures that it is unbiased and representative of the whole population, allowing for accurate inferences. Our analysis of the expected value and variance relies on this randomness to provide meaningful insights into the population's characteristics based on the sample.