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STAT151 (146)
Lecture

# Ch6.pdf

4 Pages
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School
University of Alberta
Department
Statistics
Course
STAT151
Professor
Paul Cartledge
Semester
Fall

Description
6.1 Random Variables Def’n: A random variable is a numerical measurement of the outcome of a random phenomenon. A discrete random variableis 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 6Xa0le 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, µ = x P(X = x ) ∑ i i 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, µ = ∑ x i(X = x i = (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. x P(X = x) µ = ∑ xiP(X = x i 0 0.10 1 0.05 = (0)(0.10) + (1)(0.05) + (2)(0.20) + (3)(0.65) 2 0.20 = 2.4 3 0.65 As 2 example shows, interpretation of µ as a measure of center of a distribution is more useful when the distribution is roughly symmetric, less useful when the distribution is highly skewed. The population standard deviation σ of a discrete random variable is a measure of variability of its distribution. As before, the standard deviation is defined as the square root of the population variance σ , given by 2 σ = ∑ (xi− µ) P(X = x ) i ∑ xiP(X = x ) i µ 2 Continuous Distribution: Def’n: A continuous random variable assumes any value contained in one or more intervals. e.g. average alcohol intake by a student, average alcohol outtake by a student The probability distributionof a continuous r.v. is specified by a curve. Two noticeable characteristics for continuous probability distribution (sans calculus): 1. The probability that X assumes a value in any interval lies in the range 0 to 1. 2. The interval containing all possible values has probability equal to 1, so the total area under the curve equals 1. (corresponding diagrams drawn in class) Using probability symbols, point 1 is denoted by P(a ≤ X ≤ b) = Area under the curve from a to b (diagram drawn in class) The probability that a continuous random variable X assumes a single value is always zero. This is because the area of a line, which represents a single point, is zero. (diagram drawn in class) In general, if a and b are two of the values that X can assume, then P(a) = 0 and P(b) = 0 Hence, P(a ≤ X ≤ b) = P(a < X < b). For a continuous probability distribution, the probability is always calculated for an interval. Either like abov
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