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Random Variables Probability Continued Chapter 7 Random Variables Suppose that each of three randomly selected customers purchasing a hot tub at a certain store chooses either an electric (E) or a gas (G) model. Assume that these customers make their choices independently of one another and that 40% of all customers select an electric model. The number among the three customers who purchase an electric hot tub is a random variable. What is the probability distribution? Random Variable Example X = number of people who purchase electric hot tub X 0 1 2 3 P(X) .216 .432 .288 .064 GGG (.6)(.6)(.6) EGG GEG GGE (.4)(.6)(.6) (.6)(.4)(.6) (.6)(.6)(.4) EEG GEE EGE (.4)(.4)(.6) (.6)(.4)(.4) (.4)(.6)(.4) EEE (.4)(.4)(.4) Random Variables A numerical variable whose value depends on the outcome of a chance experiment is called a random variable. discrete versus continuous Discrete vs. Continuous The number of desks in a classroom. The fuel efficiency (mpg) of an automobile. The distance that a person throws a baseball. The number of questions asked during a statistics final exam. Discrete versus Continuous Probability Distributions Which is which? Properties: For every possible x value, 0 < x < 1. Sum of all possible probabilities add to 1. Properties: Often represented by a graph or function. Area of domain is 1. Probability Histograms We can create a probability histogram to show the distributions of discrete random variables. Example Let X represent the sum of two dice. Then the probability distribution of X is as follows: X 2 3 4 5 6 7 8 9 10 11 12 P(X) 1 36 2 36 3 36 4 36 5 36 6 36 5 36 4 36 3 36 2 36 1 36 Continuous Random Variable and Density Curves The probability distribution of a continuous random variable assigns probabilities under a density curve. Probabilities are assigned to INTERVALS of outcomes rather than to individual outcomes. A probability of 0 is assigned to every individual outcome in a continuous probability distribution. The Normal Distribution can be a Probability Distribution The normal curve Means and Variances The mean value of a random variable X (written mx ) describes where the probability distribution of X is centered. We often find the mean is not a possible value of X, so it can also be referred to as the “expected value.” The standard deviation of a random variable X (written sx )describes variability in the probability distribution. Mean of a Random Variable Example Below is a distribution for number of visits to a dentist in one year. X = # of visits to the dentist. X 0 1 2 3 4 P( X ) .1 .3 .4 .15 .05 Determine the expected value, variance and standard deviation. Formulas Mean of a Random Variable m X xi pi Variance of a Random Variable s X2 ( xi m X ) 2 pi Mean of a Random Variable Example X 0 1 2 3 4 P( X ) .1 .3 .4 .15 .05 m X xi pi E(X) = 0(.1) + 1(.3) + 2(.4) + 3(.15) + 4(.05) = 1.75 visits to the dentist Variance and Standard Deviation of a Random Variable Example X 0 1 2 3 4 P( X ) .1 .3 .4 .15 .05 s ( xi m X ) pi Var(X) = (0 – 1.75)2(.1) + (1 – 1.75)2(.3) + (2 – 1.75)2(.4) + (3 – 1.75)2(.15) + = .9875 (4 – 1.75)2(.05) 2 X 2 s X .9875 .9937 visits