5Discrete_Random_Variables

5Discrete_Random_Variables - Chapter 5 Discrete Random...

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Unformatted text preview: Chapter 5 Discrete Random Variables Suppose that an experiment and a sample space are given. A random variable is a real-valued function of the outcome of the experiment. In other words, the random variable assigns a specific number to every possible outcome of the experiment. The numerical value of a particular outcome is simply called the value of the random variable. Because of the structure of real numbers, it is possible to define pertinent statistical properties on random variables that otherwise do not apply to probability spaces in general. Sample Space 1 2 3 4 5 6 7 Real Numbers Figure 5.1: The sample space in this example has seven possible outcomes. A random variable maps each of these outcomes to a real number. Example 29. There are six possible outcomes to the rolling of a fair die, namely each of the six faces. These faces map naturally to the integers one 46 5.1. PROBABILITY MASS FUNCTIONS 47 through six. The value of the random variable, in this case, is simply the number of dots that appear on the top face of the die. 1 2 3 4 5 6 Figure 5.2: This random variable takes its input from the rolling of a die and assigns to each outcome a real number that corresponds to the number of dots that appear on the top face of the die. The simplest class of random variables is the collection of discrete random variables . A variable is called discrete if its range is finite or countably infinite; that is, if the random variable can only take a finite or countable number of values. Example 30. Consider the experiment where a coin is tossed repetitively un- til heads is observed. The corresponding function, which maps the number of tosses to an integer, is a discrete random variable that takes a countable number of values. The range of this random variable is given by the positive integers { 1 , 2 , . . . } . 5.1 Probability Mass Functions A discrete random variable X is characterized by the probability of each of the elements in the range of X . We identify the probabilities of individual elements in the range of X using the probability mass function (PMF) of X , 48 CHAPTER 5. DISCRETE RANDOM VARIABLES which we denote by p X . If x is a possible value of X then the probability mass of x , written p X ( x ), is defined by p X ( x ) = Pr( { X = x } ) = Pr( X = x ) . (5.1) Equivalently, we can think of p X ( x ) as the probability of the set of all outcomes in Ω for which X is equal to x , p X ( x ) = Pr( X- 1 ( x )) = Pr( { ω ∈ Ω | X ( ω ) = x } ) . Sample Space 1 2 3 4 5 6 7 x Figure 5.3: The probability mass of x is given by the probability of the set of all outcomes which X maps to x . Let X (Ω) denote the collection of all the possible numerical values X can take; this set is known as the range of X . Using this notation, we can write x ∈ X (Ω) p X ( x ) = 1 . (5.2) We emphasize that the sets defined by { ω ∈ Ω | X ( ω ) = x } are disjoint and form a partition of the sample space Ω, as x ranges over all the possible values in X...
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5Discrete_Random_Variables - Chapter 5 Discrete Random...

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