An Introduction to Probability and Statistics, 3rd Edition

5SOME SPECIAL DISTRIBUTIONS

5.1 INTRODUCTION

In preceding chapters we studied probability distributions in general. In this chapter we will study some commonly occurring probability distributions and investigate their basic properties. The results of this chapter will be of considerable use in theoretical as well as practical applications. We begin with some discrete distributions in Section 5.2 and follow with some continuous models in Section 5.3. Section 5.4 deals with bivariate and multivariate normal distributions and in Section 5.5 we discuss the exponential family of distributions.

5.2 SOME DISCRETE DISTRIBUTIONS

In this section we study some well-known univariate and multivariate discrete distributions and describe their important properties.

5.2.1 Degenerate Distribution

The simplest distribution is that of an RV X degenerate at point k, that is, and = 0 elsewhere. If we define

(1)

the DF of the RV X is . Clearly, , , and . In particular, . This property characterizes a degenerate ...