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Random variables

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STAT 22000 Lecture Slides Random Variables Yibi Huang Department of Statistics University of Chicago Outline Coverage: Section 2.4 in the text. • Random Variables • Expected Value • Standard Deviation • Linear combinations of random variables 1 Random variables Random Variables A random variable is a numeric quantity whose value depends on the outcome of a random event • We use a capital letter, like X , to denote a random variable • The values of a random variable are denoted with a lowercase letter, in this case x, e.g., P (X = x ) Example 1. Let X be the number of heads in 2 tosses of a coin. The possible outcomes are {HH , HT , TH , TT } and X (HH ) = 2, X (HT ) = 1, X (TH ) = 1, X (TT ) = 0 Example 2. Let Y be the number of tosses required to get a head. The possible outcomes are S = {H , TH , TTH , TTTH , TTTTH , . . .} Y (H ) = 1, Y (TH ) = 2, Y (TTH ) = 3, Y (TTTH ) = 4, . . . 2 Discrete and Continuous Random Variable There are two types of random variables: • Discrete random variables often take only integer values • Example: Number of credit hours, Difference in number of credit hours this term vs last • Continuous random variables take real (decimal) values • Example: Cost of books this term, lifetime of a battery 3 Probability Distribution A probability distribution of a discrete random variable is a list of its possible values and the probabilities that it takes on those values. Value of X x1 x2 x3 Probability p1 p2 p3 ...
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