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The expected value, often just called the mean, is a key concept in statistics, representing the average outcome expected from a random variable. When dealing with the Chi-squared distribution, which arises often in statistics, especially in hypothesis testing and goodness-of-fit tests, the expected value is particularly straightforward. If a random variable \( U \) follows a Chi-squared distribution with \( u \) degrees of freedom, its expected value is simply equal to the number of degrees of freedom: \[ E(U) = u \] This makes the Chi-squared distribution unique in that its expected value is directly linked to its degrees of freedom. Remembering this relationship can be quite helpful in solving statistical problems that involve Chi-squared distributions.

Variance quantifies the spread of a set of numbers, telling us how much the values of a random variable differ from the expected value. For the Chi-squared distribution with \( u \) degrees of freedom, the variance is \[ V(U) = 2u \] This expression means that the variability of a Chi-squared distribution is twice the number of its degrees of freedom. This feature implies that as the degrees of freedom increase, not only does the expected value increase, but so does the spread of the distribution. Understanding this concept helps when determining the appropriateness of the Chi-squared test for a particular dataset, since more degrees of freedom typically involve broader data variability.

In statistics, the sample variance \( S^2 \) is a measure of how much the sample values deviate from the sample mean. It provides an estimate of the population variance \( \sigma^2 \). According to Theorem 7.3, when you have a random sample \( Y_1, Y_2, \ldots, Y_n \) from a normal distribution, the sample variance is an unbiased estimator of the population variance. This indicates that \[ E(S^2) = \sigma^2 \] This means that, on average, the sample variance gives a correct estimate of the population variance. This property is crucial in statistical inference because it assures us that the estimate is "on target" over many samples, even if any single estimate might be off. It plays a vital role in constructing confidence intervals and conducting hypothesis tests related to variance.

An unbiased estimator is a statistical term for an estimator that targets exactly the true value of a population parameter. This means that the estimation process, on average, will hit the correct parameter value. In the context of sample variance \( S^2 \), it concerns its ability to accurately estimate the true population variance \( \sigma^2 \). The expected value of the sample variance is equal to the population variance \( E(S^2) = \sigma^2 \), demonstrating that it is unbiased. When working with unbiased estimators:

• They provide trustworthy estimates across different samples.
• This reliability is fundamental for making valid inferences about the population.

Thus, in statistical practices, unbiased estimators are preferred whenever possible because they ensure that estimations don't systematically overestimate or underestimate the true parameter.