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Unformatted text preview: 6.1 Random Variables Def’n: A random variable is a numerical measurement of the outcome of a random phenomenon. A discrete random variable is a random variable that assumes separate values. e.g. # of people who think stats is dry The probability distribution of a discrete random variable lists all possible values that the random variable can assume and their corresponding probabilities. Notation: X = random variable; x = particular value; P ( X = x ) denotes probability that X equals the value x . Ex6.1) Toss a coin 3 times. Let X be the number of heads. 8 possible values: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT Table 6X0 x P ( X = x ) 0 0.125 1 0.375 2 0.375 3 0.125 Two noticeable characteristics for discrete probability distribution: 1. 0 ≤ P ( X = x ) ≤ 1 for each value of x 2. ∑ P ( X = x ) = 1 Ex6.2) “no heads”: P ( X = 0) = 0.125 “at least one head”: P ( X ≥ 1) = P ( X = 1) + P ( X = 2) + P ( X = 3) = 0.375 + 0.375 + 0.125 = 0.875 “less than 2 heads”: P ( X < 2) = P ( X = 0) + P ( X = 1) = 0.125 + 0.375 = 0.500 The population mean μ of a discrete random variable is a measure of the center of its distribution. It can be seen as a long-run average under replication. More precisely, ∑ = = ) ( i i x X P x µ Sometimes referred to as μ = E ( X ) = the expected value of X (see STAT 265). Keep in mind that μ is not necessarily a “typical” value of X (it’s not the mode). Ex6.3) Using Table 6X0, ∑ = = ) ( i i x X P x µ = (0)(0.125) + (1)(0.375) + (2)(0.375) + (3)(0.125) = 1.5 Æ On average, the number of heads from 3 coin tosses is 1.5. Ex6.4) Toss an unfair coin 3 times (hypothetical). Let X be as in previous example....
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- Winter '07