Random Variables Probability Continued Chapter 6 Random Variables

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) EGG GEG GGE (. 4)(. 6)(. 4) EEG GEE EGE (. 4)(. 6)(. 4) EEE (. 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. l Sum of all possible probabilities add to 1. l Continuous Properties: Often represented by a graph or function. l Can take on any value in an interval. l Area of domain is 1. l

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 E(X) = 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?

2 X What is the probability distribution for 2 X?

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(3 X) = ? , Var(3 X) = ? E(3 X) = 3*2. 5 = 7. 5 Var(3 X) = 32 * 0. 2 = 1. 8 What is E(2 X – 1) = ? , Var(2 X – 1) = ? E(2 X – 1) = 2(2. 5) – 1 = 4 Var(2 X – 1) = 22 * 0. 2 = 0. 8

Transforming Rules If X is a random variable and a and b are fixed numbers, then ma + b. X = a + bm. X If X is a random variable and a and b are fixed numbers, then s 2 a + b. X =b 2 s 2 X