• A study of education followed a large group of fourth-grade children to see how many years of school they eventually completed. Let x be the highest year of school that a randomly chosen fourth grader completes. (Students who go on to college are included in the outcome x = 12.) The study found this probability distribution for x.– What percent of fourth graders eventually finished

twelfth grade?– Check that this is a legitimate discrete probability

distribution.– Find P(X≥6). – Find P(X>6).– What values of x make up the event “the students

completed at least one year of high school”? (High school begins with ninth grade.) What is the probability of this?Years 4 5 6 7 8 9 10 11 12Prob. .01

0.007

.007

.013

.032 .068 .070 .041

.752

Section 6.2Section 6.2Section 6.2Section 6.2

Means and Variances of Random Means and Variances of Random VariablesVariables

Mean of a Random Variable

• In the Tri-State Pick 3 Lottery game, you choose a 3-digit number. The state chooses a 3-digit winning number at random and awards you $500 if your number is chosen.

• Construct a probability distribution for the random variable X, the payoff amount of the lottery game. We will come back to this in just a minute!

μ vs. x-bar• The mean of a probability distribution

describes the long-run average outcome. It is the idealized mean, so we denote it with the Greek letter mu, μ.

• On the other hand, you could play the Pick 3 game a few times and calculate the mean of the actual amounts you win. That would be called x-bar.

• The mean of a random variable X is often called the expected value of X. This name is misleading, because sometimes the mean isn’t even a possible value for X.

How to Find the Mean of a Discrete R.V.

• Finding the mean of a discrete random variable is simply a weighted average.

• If the distribution of X is as shown in the table, then the mean of X is

Value of X x1 x2 x3 … xk Probability

p1 p2 p3 … pk

x i ix p

In other words… the mean is the sum of each outcome times the probability of that outcome

The x is to denote which variable we are looking at the mean of.

Average Payoff• What is the average payoff of the

Pick 3 game for many tickets?– Note: This is not a possible outcome.

• If tickets cost $1, in the long run how much does the state keep of the money you wager?

• We would say this is a “fair game” if the state made $0 profit.

Find the MeanUnits Sold 1000 3000 5000 10,000

Probability 0.1 0.3 0.4 0.2

Variance of a Discrete Random Variable

• Here is the formula for the variance of a discrete random variable:

• In other words, the variance is the sum of the squares of the differences between the outcome and the mean times the probability of that outcome.

2 2( )x i x ix p

Remember the Relationship between Variance and

Standard Deviation

• Standard Deviation is the square root of the variance.

• To find the standard deviation, just take the square root of the variance.

Find the Standard Deviation

Units Sold 1000 3000 5000 10,000

Probability 0.1 0.3 0.4 0.2

Homework:Homework:Worksheet on Worksheet on

Random VariablesRandom Variables

Homework:Homework:Worksheet on Worksheet on

Random VariablesRandom Variables