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Random Variables

Random Variables. Probability Continued Chapter 7. Random Variables.

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Random Variables

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  1. Random Variables Probability Continued Chapter 7

  2. 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?

  3. Random Variable Example X = number of people who purchase electric hot tub X 0 1 2 3 .288 P(X) .216 .432 .064 (.6)(.6)(.6) GGG EEG GEE EGE (.4)(.4)(.6) (.6)(.4)(.4) (.4)(.6)(.4) EGG GEG GGE (.4)(.6)(.6) (.6)(.4)(.6) (.6)(.6)(.4) EEE (.4)(.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

  5. 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.

  6. 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.

  7. Probability Histograms • We can create a probability histogram to show the distributions of discrete random variables.

  8. Example • Let X represent the sum of two dice. •  Then the probability distribution of X is as follows:

  9. 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.

  10. The Normal Distribution can be a Probability Distribution • The normal curve

  11. 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.

  12. 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. • Determine the expected value, variance and standard deviation.

  13. Formulas Mean of a Random Variable Variance of a Random Variable

  14. Mean of a Random Variable Example E(X) = 0(.1) + 1(.3) + 2(.4) + 3(.15) + 4(.05) = 1.75 visits to the dentist

  15. Variance and Standard Deviation of a Random Variable Example Var(X) = (0 – 1.75)2(.1) + (1 – 1.75)2(.3) + (2 – 1.75)2(.4) + (3 – 1.75)2(.15) + (4 – 1.75)2(.05) = .9875

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