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Unformatted text preview: NOTES ON RANDOM VARIABLES AND THEIR DISTRIBUTIONS by A. Ledoan In the first three chapters of his book  Ross sets forth the formulation of prob- ability theory. The idea of an experiment, a sample space corresponding to the experiment, and events in a sample space were introduced and the axioms of a probability measure on these events were postulated. In Chapter 4 Ross uses the concept of a random variable to unify the study of probabilistic situations. This is achieved by mapping the original sample space to the real line R , for any experi- ment. Thus we will only have to study the one sample space, R . There are basically two distinct types of random variables, discrete and continuous. In Chapter 4 the emphasis will be on discrete random variables while much of the remainder of the textbook will deal with problems associated with a continuous random variable. 1. The Concept of a Random Variable Let S be a sample space associated with a random experiment. Then each point s ∈ S specifies an outcome. If X is a random variable, each outcome will specify the value of X . So associated with each s ∈ S there is a real number X ( s ). Definition 1. Let S be a sample space. A random variable X is a function that assigns a real number X ( s ) to each element s ∈ S . This means that X : S → R . The domain of X is S and the range of X is R or a subset of R , depending on what possible values the random variable can take. The first thing to note about this definition is that X is not really a variable at all. It is a function that maps elements of S to R . What is more, it is also not randome in the sense that once the outcome s ∈ S is known, the real number X ( s ) is completely determine. Rather it is the outcome s ∈ S that is random, not the mapping. Definition 2. A discrete random variable is a random variable whose range is finite or countable. This means that the values of such a random variable can be listed, or arranged in a sequence. Nothing in the definition requires that discrete random variables be integers. The collection of all the probabilities related to a discrete random variable X is the distribution of X . By the distribution of X we mean the entire information about the behavior of X . The function p X ( x ) = P ( X = x ) , for all x , is the probability mass function (for short, pmf) of X . Since p X ( x ) is a proba- bility, it satisfies p X ( x ) ∈ [0 , 1] , for all x . Moreover, for every outcome s ∈ S , the random variable takes exactly one value x . This makes the event X = x disjoint and exhaustive, and so and X x p X ( x ) = 1 . The cummulative distribution function (for short, cdf) of X is defined as F X ( x ) = P ( X ≤ x ) = X u ≤ x p X ( u ) . 2 Notes on Random Variables and Their Distributions This last formula tells us that F X ( x ) is a monotonic non-decreasing function of X , always between 0 and 1, with lim x →∞ F X ( x ) = 1 and lim x →-∞ F X ( x ) = 0 ....
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