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4.1 Probability Distributions • Important Concepts – Random Variables – Probability Distribution – Mean (or Expected Value) of a Random Variable – Variance and Standard Deviation of a Random Variable 4.1 Probability Distributions • Consider the following experiment: – Suppose we toss a coin three times. • What is the sample space for this experiment? • What is the probability of tossing exactly: 0 tails? 1 tails? 2 tails? 3 tails? 4.1 Probability Distributions • Terms to know: – A random variable X represents a numerical value associated with each outcome of a probability experiment. • A random variable is discrete if it has a finite or countable number of possible outcomes. • A random variable is continuous if it has an uncountable number of possible outcomes. 4.1 Probability Distributions • Terms to know: – A discrete probability distribution lists each possible value a random variable can assume, together with its probability. • All probability distributions must satisfy the following two conditions: – The probability of each value of the random variable must be between 0 and 1, inclusive. – The sum of all the probabilities must be 1. #26 p. 198 #28 p. 198 4.1 Probability Distributions • Discrete probability distribution of our random variable X: # of tails, x P( X = x ) 0 1/8 = 0.125 1 3/8 = 0.375 2 3/8 = 0.375 3 1/8 = 0.125 1.000 4.1 Probability Distributions • How do we find the mean and standard deviation of a discrete random variable? In chapter 3, we used the following: x N x 2 x 2 N N Can we still use these formulas? 2 2 x N 2 x N 2 2 4.1 Probability Distributions • Let’s try #31 p. 199 (Camping Chairs) Number of Cats, x P(X=x) x·P(X=x) x2 x2·P(X=x) 0 0.250 0 0 0 1 0.298 0.298 1 0.298 2 0.229 0.458 4 0.916 3 0.168 0.504 9 1.512 4 0.034 0.136 16 0.544 5 0.021 0.105 25 0.525 1.501 3.795 4.1 Probability Distributions • #33 p. 199 (Hurricanes) Category, x P(X = x) x∙P(X =x) x2 x2∙P(X =x) 1 0.418 0.418 1 0.418 2 0.261 0.522 4 1.044 3 0.247 0.741 9 2.223 4 0.063 0.252 16 1.008 5 0.010 0.05 25 0.25 1.983 4.943