By L. Sucheston
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Extra resources for Contributions to Ergodic Theory and Probability
In addition, we will provide examples of some important and frequently encountered random variables. In Chapter 3, we will discuss general (not necessarily discrete) random variables. Even though this chapter may appear to be covering a lot of new ground, this is not really the case. ) and apply them to random variables rather than events, together with some appropriate new notation. The only genuinely new concepts relate to means and variances. 2 PROBABILITY MASS FUNCTIONS The most important way to characterize a random variable is through the probabilities of the values that it can take.
We have illustrated through examples three methods of specifying probability laws in probabilistic models: (1) The counting method. This method applies to the case where the number of possible outcomes is ﬁnite, and all outcomes are equally likely. To calculate the probability of an event, we count the number of elements in the event and divide by the number of elements of the sample space. (2) The sequential method. This method applies when the experiment has a sequential character, and suitable conditional probabilities are speciﬁed or calculated along the branches of the corresponding tree (perhaps using the counting method).
What is the probability that each group includes a graduate student? 3, but we will now obtain the answer using a counting argument. We ﬁrst determine the nature of the sample space. A typical outcome is a particular way of partitioning the 16 students into four groups of 4. We take the term “randomly” to mean that every possible partition is equally likely, so that the probability question can be reduced to one of counting. According to our earlier discussion, there are 16 4, 4, 4, 4 = 16! 4! 4!
Contributions to Ergodic Theory and Probability by L. Sucheston