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05-randvars-discr-1 - IE 111 Fall Semester 2011 Note Set#5...

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IE 111 Fall Semester 2011 Note Set #5 Random Variables Random Variables The concepts of random variables and probability distributions are extremely important in probability and statistics. Once you see clearly in your mind what they are, and how they work, you will be in good shape. They are really quite simple. Here are a couple of examples of random variables: Example 1. Expt: Roll 2 dice Let X be a random variable. Let X = the sum of the two dice. Example 2. Expt: Toss 10 coins Let Y be a random variable where Y = the number of “heads” Example 3. Expt: Randomly select a student from class. Define the following random variables: X = the height in inches of the student Y = the weight in pounds of the student Z = 0 if the student is a male or Z=1 if the student is a female. Implicit in the definition of a particular random variable are an experiment , and outcomes mapped to numbers . From the examples we start to see some properties of random variables. Properties of Random Variables Property 1 After an experiment is conducted and the outcome is observed, the random variable takes on a numerical value. The outcome of a random variable is a number. Example Suppose we select a random student from class and let X=“Male” if the student is male and let X=“female” if the student is female. Is X a random variable? NO . since “male” and “female” are not numbers.
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Property 2. All random variables (X) have an associated “domain” which is the set of all possible outcomes for X. Example. In Example 1 above where 2 dice are rolled and X= their sum, the domain of X is {1,2,3,...,11,12} right? Wrong…Not 1 When Y = the number of heads in 10 flips, the domain is {0, 1, 2,..., 9, 10} When the height and weight of a randomly selected student is observed, the domains of both X and Y could be all positive real numbers. When the student's sex is observed and mapped to Z = 0 or 1, the domain of Z is {0,1} Property 3 X has a probability distribution function which assigns a probability to each element of the domain. Probability Distributions are the most important concept to understand in Prob and Stats. There can be some confusion in both terminology and notation here. The terms “Probability Distribution”, “Probability Mass Function”, and “Probability Density Function” are often used interchangeably. The book tends to use the following notation: Let f(x) be the probability distribution function of random variable X. I will try to use the more precise notation and definitions: Discrete Random Variables are random variables with finite or countable domains. Continuous Random Variables have, basically, domains that are intervals of real numbers. We will study these things in the next chapter. Discrete Random Variables have Probability Mass Functions , denoted by f X (x) for random variable X, f Y (y) for random variable Y, etc.
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