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

XP(X)

X = number of people who purchase electric hot tub

0 1 2 3

GGG (.6)(.6)(.6)

.216

EGGGEGGGE

(.4)(.6)(.6)(.6)(.4)(.6)(.6)(.6)(.4)

.432

EEGGEEEGE

(.4)(.4)(.6)(.6)(.4)(.4)(.4)(.6)(.4)

.288

EEE (.4)(.4)(.4)

.064

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 12P(X) 1

36236

3 36

436

536

636

536

436

336

236

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 DistributionThe normal curve

Means and VariancesThe 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.

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.

0 1 2 3 4

( ) .1 .3 .4 .15 .05

X

P X

Formulas

Mean of a Random Variable

X i ix p

Variance of a Random Variable

2 2( )X i X ix p

Mean of a Random Variable Example

0 1 2 3 4

( ) .1 .3 .4 .15 .05

X

P X

E(X) = 0(.1) + 1(.3) + 2(.4) + 3(.15) + 4(.05)

= 1.75 visits to the dentist

X i ix p

Variance and Standard Deviation of a Random Variable Example

0 1 2 3 4

( ) .1 .3 .4 .15 .05

X

P X

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

2 2( )X i X ix p

.9875 .9937 visitsX