Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers, 3rd Edition

10

The Sample Mean

Earlier chapters of this book present the properties of probability models. In referring to applications of probability theory, we have assumed prior knowledge of the probability model that governs the outcomes of an experiment. In practice, however, we encounter many situations in which the probability model is not known in advance and experimenters collect data in order to learn about the model. In doing so, they apply principles of statistical inference, a body of knowledge that governs the use of measurements to discover the properties of a probability model.

This chapter focuses on the properties of the sample mean of a set of data. We refer to independent trials of one experiment, with each trial producing one sample value of a random variable. The sample mean is simply the sum of the sample values divided by the number of trials. We begin by describing the relationship of the sample mean of the data to the expected value of the random variable. We then describe methods of using the sample mean to estimate the expected value.

10.1   Sample Mean: Expected Value and Variance

The sample mean M_n(X) = (X₁ + … + X_n)/n of n independent observations of random variable X is a random variable with expected value E[X] and variance Var[X]/n.

In this section, we define the sample mean of a random variable and identify its expected value and variance. Later sections of this chapter show mathematically how the sample mean converges to a constant as the number of repetitions ...