Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

3.7 PARAMETRIC MODELS FOR RANDOM VARIABLES

In many applications where the data are random, we are interested in deriving a distribution model that accurately represents the observed samples. The modeling process has the following components:

• A parametric family that “best” represents the observed samples is chosen as a model for the distribution of random variable X. For example, from a histogram of observed samples, we might decide that a Gaussian distribution with pdf FX(x) is a reasonable model because the shape of the histogram is “bell-shaped.”
• From observed samples of X, estimate the parameters of the distribution. For the Gaussian example, the sample mean can be used to estimate the mean , and the sample variance S2 for the variance . Several parameter-estimation techniques are discussed in Chapter 9.
• Once a parametric model has been chosen, various characteristics of the random variable can be determined, such as specific probabilities or moments like skewness and kurtosis. Moments and other expectations are described in Chapter 5.
• The parametric model can be used to predict future outcomes that might be used to make decisions in various applications, such as those mentioned ...