2. Random Variables Probability Continued Chapter 7

3. 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?

4. Random Variable Example X P(X) X = number of people who purchase electric hot tub 01 23 GGG (.6)(.6)(.6).216 EGG GEG GGE (.4)(.6)(.6) (.6)(.4)(.6) (.6)(.6)(.4).432 EEG GEE EGE (.4)(.4)(.6) (.6)(.4)(.4) (.4)(.6)(.4).288 EEE(.4)(.4)(.4).064

5. Random Variables A numerical variable whose value depends on the outcome of a chance experiment is called a random variable. discrete versus continuous

6. 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.

7. 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.

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

9. Example Let X represent the sum of two dice. Then the probability distribution of X is as follows: X23456789101112 P(X) 1 36 2 36 3 36 4 36 5 36 6 36 5 36 4 36 3 36 2 36 1 36

10. 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.

11. The Normal Distribution can be a Probability Distribution The normal curve

12. Means and Variances The mean value of a random variable X (written  x ) 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  x )describes variability in the probability distribution.

13. 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.

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

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

16. 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