Chapter 7 Random Variables

7.1: Discrete and Continuous Random Variables

Random Variables

• A random variable is a variable whose value is a numerical outcome of a random phenomenon. – the basic units of sampling distributions.– 2 types: discrete and continuous

Discrete Random Variables

• A discrete random variable X has a countable number of possible values.

• The probability distribution of a discrete random variable X lists the values and their probabilities.

Value of X: x1 x2 x3 … xk

Probability:

p1 p2 p3 … pk

Probability Distribution

• The probability pi must satisfy two requirements.– Every probability pi is an number between 0

and 1– The sum of the probabilities is 1:

p1 + p2 + p3 +…+pk = 1

• Find the probability of any event by adding the probabilities pi of the particular values xi that make up the event.

Example: Maturation of male college students

• In an article in the journal Developmental Psychology (March 1986), a probability distribution for the age X (in years) when male college students began to shave regularly is shown:

X 11 12 13 14 15

P(X) 0.013 0 0.027 0.067 0.213

X 16 17 18 19 ≥20

P(X) 0.267 0.240 0.093 0.067 0.013

Example

• Page 470 #7.2

Example

• Page 470 #7.4

Continuous Random Variables

• A continuous random variable X takes on all values in an interval of numbers.

• The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up that event.

• All continuous probability distributions assign probability 0 to every individual outcome.

Example: Violence in Schools

• Page 476 #7.9

Example: Drugs in schools

• An opinion poll asks a SRS of 1500 American adults what they consider to be the most serious problem facing our schools. Suppose that if we could ask all adults this question, 30% would say “drugs”. What is the probability that the poll result differs from the truth about the population by more than two percentage points? N(.3, 0.0118)

Chapter 7 Random Variables

7.2: Means and Variances of Random Variables

Activity 7B

• Page 481

Mean and expected Value

• Mean of a probability distribution is denoted by µ, or µx.

• The mean of the random variable, X is often referred to as the expected value of X.

Mean of a Discrete Random Variable

• Suppose that X is a discrete random variable whose distribution is

To find the mean of X, multiply each possible value by its probability, then add all the products.

Value of X: x1 x2 x3 … xk

Probability:

p1 p2 p3 … pk

Example

• Page 486 #7.24

Example

• Using the data from the “Maturation of male college students” example, find and interpret the mean.

Variance of a Discrete Random Variable

• Suppose that X is a discrete random variable whose distribution is

and that µ is the mean of X. The variance of X is

σx2 = Σ(x1 - µx)2pi.

The standard deviation σx of X is the square root of the variance.

Value of X: x1 x2 x3 … xk

Probability:

p1 p2 p3 … pk

Example

• Page 486 #7.28

Technology Tip

• To find µx and σx:

Example

• Using the data from the “Maturation of male college students” example, find the standard deviation.

• Use the empirical rule to determine if the “Maturation of male college students” data is normally distributed.

Sampling Distributions

• The sampling distributions of statistics are just the probability distributions of these random variables.

Law of Large Numbers

• The average of a randomly selected sample from a large population is likely to be close to the average of the whole population.

Law of Large Numbers

• What is the mean of rolling 3 dice?

Example:Emergency Evacuations

• A panel of meteorological and civil engineers studying emergency evacuation plans for Florida’s Gulf Coast in the event of a hurricane has estimated that it take between 13 and 18 hours to evacuate people living in low-lying land, with the probabilities shown in the table.

Let X = the time it takes a randomly selected person living in low-lying land in Florida to evacuate.

Time to Evacuate Probability

(nearest hour)

13 0.04

14 0.25

15 0.40

16 0.18

17 0.10

18 0.03

• Is X a discrete random variable or a continuous random variable?

• Sketch a probability histogram for this data.

• Find and interpret the mean.

• Find the standard deviation.

• Weather forecasters say that they cannot accurately predict a hurricane landfall more than 14 hours in advance.

Find the probability that all residents of low-lying areas are evacuated safely if the Gulf Coast Civil Engineering Department waits until the 14-hour warning before beginning evacuation.

Law of Small Numbers

• Gambler’s Fallacy is the belief that every segment of a random sequence should reflect the true proportion.

• This is a myth. There is no law of small numbers!

Rules for Means

• Rule 1: If X is a random variable and a and b are fixed numbers, then

μa+bx = a + bμx

• Rule 2: If X and Y are random variables, then

μX + Y = μX + μY

Rules for Variances

• Rule 1: If X is a random variable and a and b are fixed numbers, then

σ2a+bx = b2σ2

x

• Rule 2: If X and Y are independent random variables, then

σ2X + Y = σ2

x + σ2Y

σ2X - Y = σ2

x + σ2Y

Means and Variances

• Variances add, standard deviations don’t.

• These rules can extend for more than 2 random variables…just follow the pattern.

Combining Normal Random Variables

• Any linear combination of independent Normal random variables is also Normally distributed.

Example

• Page 499 #7.38

• Page 499 #7.40

• Page 500 #7.42

Example

• Page 501 #7.44

Example

• Page 503 #7.50

Example

• Page 503 #7.52