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 12 Prob. .010 .007 .013 .032 .068 .070 .041 .752

Means and Variances of Random Variables Section 6.2 Means and Variances of Random Variables

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 The x is to denote which variable we are looking at the mean of. Value of X x1 x2 x3 … xk Probability p1 p2 p3 pk In other words… the mean is the sum of each outcome times the probability of that outcome

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 Mean Units 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.

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: Worksheet on Random Variables