Probability Continued Chapter 6 Random Variables Probability Continued Chapter 6

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) 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)

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 Discrete Properties: For every possible x value, 0 < p < 1. Sum of all possible probabilities add to 1. Continuous Properties: Often represented by a graph or function. Can take on any value in an interval. Area of domain is 1.

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. Determine the expected value, variance and standard deviation.

Formulas Mean of a Random Variable Variance of a Random Variable

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

Variance and Standard Deviation 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

Developing Transformation Rules Consider the following distribution for the random variable X:

X+1 What is the probability distribution for X+1?

2X What is the probability distribution for 2X?

Consider Suppose that E(X) = 2.5, Var(X) = 0.2 What is E(X+5) = ?, Var(X+5) = ? E(X+5) = 2.5 + 5 = 7.5 Var(X+5) = 0.2 (no change) What is E(X – 2.2) = ?, Var(X – 2.2) = ? E(X – 2.2) = 2.5 – 2.2 = 0.3 Var(X – 2.2) = 0.2 (no change)

Consider Suppose that E(X) = 2.5, Var(X) = 0.2 What is E(3X) = ?, Var(3X) = ? E(3X) = 3*2.5 = 7.5 Var(3X) = 32 * 0.2 = 1.8 What is E(2X – 1) = ?, Var(2X – 1) = ? E(2X – 1) = 2(2.5) – 1 = 4 Var(2X – 1) = 22 * 0.2 = 0.8

Transforming Rules If X is a random variable and a and b are fixed numbers, then ma + bX = a + bmX If X is a random variable and a and b are fixed numbers, then s2a + bX =b2s2X