Question

Find the expected value of the random variable \(\mathrm{S}^{2}{*}=(1 / \mathrm{n})^{\mathrm{n}} \sum_{\mathrm{i}=1}\left(\mathrm{X}_{\mathrm{i}}-\underline{\mathrm{X}}\right)^{2}\), where \(\underline{\mathrm{X}}={ }^{\mathrm{n}} \sum_{\mathrm{i}=1} \mathrm{X}_{\mathrm{i}} / \mathrm{n}\) and the \(\mathrm{X}_{\mathrm{i}}\) are independent and identically distributed with \(\mathrm{E}\left(\mathrm{X}_{\mathrm{i}}\right)=\mu\), Var \(\mathrm{X}_{\mathrm{i}}=\sigma^{2}\) for \(\mathrm{i}=1,2 \ldots \ldots \mathrm{n}\)

Short answer

The short answer based on the given step-by-step solution is: The expected value of \(S^{2}{*}\) is given by: \[ E(S^{2}{*}) = \sigma^2 + \frac{1}{n}\sigma^2 \]

Step-by-step solution

Recall properties of expected value

Recall the following properties of expected values: 1. Linearity property: \(E(aX + b) = aE(X) + b\) for any constants \(a\) and \(b\). 2. Independence property: \(E(XY) = E(X)E(Y)\) if \(X\) and \(Y\) are independent random variables.

Expand the expression for \(S^{2}{*}\)

Let's first expand \(S^{2}{*}\): \[S^{2}{*} = (1/n)^n \sum_{i=1}^{n}\left(X_i - \frac{1}{n} \sum_{j=1}^{n} X_j\right)^{2}\]

Find the expected value of \(S^{2}{*}\) using the properties of expected values

Using the linearity property of expected values, we have: \[E(S^{2}{*}) = (1/n)^n \sum_{i=1}^n E\left(\left(X_i - \frac{1}{n} \sum_{j=1}^n X_j\right)^2\right)\] Now, let's find the expectation of the inner expression: \begin{align*} E\left(\left(X_i - \frac{1}{n} \sum_{j=1}^n X_j\right)^2\right) &= E\left(X_i^2 - 2X_i \frac{1}{n}\sum_{j=1}^n X_j + \frac{1}{n^2} \left(\sum_{j=1}^n X_j\right)^2\right) \\ &= E\left(X_i^2\right) - 2E\left(X_i \frac{1}{n}\sum_{j=1}^n X_j\right) + E\left(\frac{1}{n^2} \left(\sum_{j=1}^n X_j\right)^2\right) \end{align*} Let's calculate each term separately. For the first term, we have: \(E(X_i^2) = \sigma^2 + \mu^2\), since Var(\(X_i\)) = \(E\left(X_i^2\right) - \mu^2\). For the second term, we have: \begin{align*} E\left(X_i \frac{1}{n}\sum_{j=1}^n X_j\right) &= \frac{1}{n}E(X_i)\sum_{j=1}^n E(X_j) = \frac{1}{n}\mu^2 n = \mu^2 \end{align*} For the third term, we have: \begin{align*} E\left(\frac{1}{n^2} \left(\sum_{j=1}^n X_j\right)^2\right) &= \frac{1}{n^2} E\left(\left(\sum_{j=1}^n X_j\right)^2\right) \\ &= \frac{1}{n^2} E\left(\sum_{j=1}^n \sum_{k=1}^n X_j X_k\right) \end{align*} Applying the fact that \(E(X_jX_k) = E(X_j)E(X_k) = \mu^2\) (since \(X_j\) and \(X_k\) are independent) for \(j\neq k\), we get: \[\frac{1}{n^2} E\left(\sum_{j=1}^n X_j^2 + \sum_{j\neq k} X_j X_k\right) = \frac{1}{n^2}(n(\sigma^2 + \mu^2) + n(n-1)\mu^2)\] Combining all three terms, we get: \begin{align*} E\left(\left(X_i - \frac{1}{n} \sum_{j=1}^n X_j\right)^2\right) &= (\sigma^2 + \mu^2) - 2\mu^2 + \frac{1}{n^2}(n(\sigma^2 + \mu^2) + n(n-1)\mu^2) \\ &= \sigma^2 + (1-n)\mu^2 + \frac{1}{n}\sigma^2 + \frac{n-1}{n}\mu^2 \\ &= \sigma^2 + \frac{1}{n}\sigma^2 \end{align*}

Substitute the result back in and simplify

Substituting this result back into the expression for \(E(S^{2}{*})\), we get: \begin{align*} E(S^{2}{*}) &= (1/n)^n \sum_{i=1}^n \left(\sigma^2 + \frac{1}{n}\sigma^2\right) \\ &= (1/n)^n n \left(\sigma^2 + \frac{1}{n}\sigma^2\right) \\ &= \sigma^2 + \frac{1}{n}\sigma^2 \end{align*} Thus, the expected value of \(S^{2}{*}\) is: \[ E(S^{2}{*}) = \sigma^2 + \frac{1}{n}\sigma^2 \]

Key concepts

Random Variables

A random variable is a mathematical function that represents a possible outcome of a random phenomenon. In statistics, random variables are used to map the outcomes of random events to numbers. This concept helps in performing mathematical operations and analyzing probabilities. For example, if you're rolling a die, the possible outcomes from one roll can be mapped to the numbers from 1 to 6, and each number is a random variable.
Independent random variables are those whose occurrence is not affected by other variables. This is fundamental in statistical analysis as it allows us to study variables separately without interference. Understanding random variables is crucial in analyzing and predicting patterns and trends in data.

Variance and Standard Deviation

Variance and standard deviation are statistical tools used to measure the spread or dispersion of a set of data.
Variance tells us how much the values in a data set diverge from the mean of the data. A high variance indicates that the data points are spread out over a large range of values, while a low variance indicates they are closer to the mean.

• Variance (\( ext{Var}(X) \)) is calculated as the average of the squared differences from the mean.
• Standard deviation (\( ext{SD}(X) \)) is the square root of the variance and provides a measure of spread in the same units as the data.

These metrics are applicable almost every time we work with statistics, providing insights into the reliability and variability of our data.

Linearity of Expectation

The linearity of expectation is a useful property in probability and statistics, stating that the expected value of the sum of random variables is equal to the sum of their expected values.
Mathematically, if \( X_1, X_2, ..., X_n \) are random variables, then:

• \( E(X_1 + X_2 + ... + X_n) = E(X_1) + E(X_2) + ... + E(X_n) \)

This property holds regardless of whether the random variables are independent, which simplifies calculations by allowing us to focus on finding the expected values of individual variables and summing them up. In practice, this means if you know the expected values of a few separate experiments, you can easily find the expected value of the whole system.

Independent and Identically Distributed (i.i.d) Variables

Independent and identically distributed (i.i.d) variables are a group of variables that have two specific properties: independence and identical distribution.

• Independence means that the occurrence or result of one variable does not affect the others.
• Identical distribution indicates that each variable has the same probability distribution.

This concept is significant in the realm of probability because i.i.d variables simplify calculations and prediction models. Many statistical models assume i.i.d variables because they remove complexity associated with correlated data. For instance, when each variable represents the outcome of rolling a fair die, all are i.i.d as each die roll does not impact another, and all have the same outcome probability.