376 Part IV • Randomness and Probability

Continuous Random Variables

A S Numeric Outcomes.You've seen how to simulate con-tinuous random outcomes. It turnsout there's a tool for simulatingcontinuous outcomes, too. See ithere.

A S Means of RandomVariables. Experiment with contin-uous random variables to learnhow their expected values behave.

A company manufactures small stereo systems. At the end of the production line,the stereos are packaged and prepared for shipping. Stage 1 of this process iscalled "packing." Workers must collect all the system components (a main unit,two speakers, a power cord, an antenna, and some wires), put each in plasticbags, and then place everything inside a protective styrofoam form. The packedform then moves on to Stage 2, called "boxing." There, workers place the formand a packet of instructions in a cardboard box, close it, then seal and label thebox for shipping.

The company says that times required for the packing stage can be described bya Normal model with a mean of 9minutes and standard deviation of 1.5minutes.The times for the boxing stage can also be modeled as Normal, with a mean of 6minutes and standard deviation of 1minute.

This is a common way to model events. Do our rules for random variablesapply here? What's different? We no longer have a list of discrete outcomes, withtheir associated probabilities. Instead, we have continuous random variables thatcan take on any value. Now any single value won't have a probability. We sawthis back in Chapter 6 when we first saw the Normal model (although we didn'ttalk then about "random variables" or "probability"). We know that the probabil-ity that z = 1.5doesn't make sense, but we can talk about the probability that z liesbetween 0.5and 1.5.For a Normal random variable (and any other continuous ran-dom variable) the probability that it falls within an interval is just the area underthe curve over that interval.

Some continuous random variables have Normal models; others may be skewed,uniform, or bimodal. Regardless of shape, all continuous random variables havemeans (which we also call expected values) and variances. In this book we won'tworry about how to calculate them, but we can still work with models for contin-uous random variables when we're given these parameters.

The good news is that nearly everything we've said about how discrete randomvariables behave is true of continuous random variables, as well. When two inde-pendent continuous random variables have Normal models, so does their sum ordifference. This simple fact is a special property of Normal models and is very im-portant. It allows us to apply our knowledge of Normal probabilities to questionsabout the sum or difference of independent random variables.

Packaging Stereos Step-By-Step

Consider the company that manufactures and ships small stereo systems that wediscussed above.

Recall that times required to pack the stereos can be described by a Normalmodel with a mean of 9minutes and standard deviation of 1.5minutes. The timesfor the boxing stage can also be modeled as Normal, with a mean of 6 minutesand standard deviation of 1minute.

Two questions:

1. What is the probability that packing two consecutive systems takes over 20minutes?2. What percentage of the stereo systems take longer to pack than to box?

\

Chapter 16 • Random Variables 377

Plan State the problem.

Question 1: What is the probability that packing two consecutive systems takes over 20 minutes?

Variables Define your random variables.

Write an appropriate equation.

Check the conditions. Sums of independentNormal random variables follow a Normalmodel. Such simplicity isn't true in general.

Mechanics Find the expected value.

For sums of independent random variables,variances add. (We don't need the variables tobe Normal for this to be true-just indepen-dent.)

Find the standard deviation.

Now we use the fact that both random vari-ables follow Normal models to say that theirsum is also Normal.

Sketch a picture of the Normal model for thetotal time, shading the region representingover 20 minutes.

Find the z-score for 20 minutes.Use technology or a table to find theprobability.

Conclusion Interpret your result in context.

I want to estimate the probability that packingtwo consecutive systems takes over 20 minutes.

Let P1 = time for packing the first systemP2 = time for packing the secondT = total time to pack two systems

T= P1 + P2

V Normal models: Weare told that both ran-dom variables follow Normal models.

V Independence: Wecan reasonably assumethat the two packing times are independent.

E(1) = E(P1 + P2)

= E(P1) + E(P2)

= 9 + 9 = 18 minutesSince the times are independent,Var(1) = Var(P1 + P2)

= Var(P1) + Var(P2)

= 1.52 + 1.52Var(1) = 4.50SD(1) = V4.50 = 2.12 minutesI'll model Twith N (18, 2.12).

18 20

20 - 18z = = 0.94

2.12

P(T> 20) = P(z> 0.94) = 0.1736There's a little more than a 17/0 chance that itwill take a total of over 20 minutes to pack twoconsecutive stereo systems.

Question 2: What percentage of stereo systems take longer to pack than to box?

Plan State the question.

Variables Define your random variables.

Write an appropriate equation.

I want to estimate the percentage of the stereosystems that take longer to pack than to box.

Let P = time for packing a systemB = time for boxing a systemD = difference in times to pack and boxa

systemD=P-B

378 Part IV • Randomness and Probability

What are we trying to find? Notice that wecan tell which of two quantities is greater bysubtracting and asking whether the differenceis positive or negative.

Don't forget to check the conditions.

Mechanics Find the expected value.

For the difference of independent randomvariables, variances add.

Find the standard deviation.

State what model you will use.

Sketch a picture of the Normal model for thedifference in times and shade the region rep-resenting a difference greater than zero.

Find the z-score for a minutes, then use atable or technology to find the probability.

Conclusion Intepret your result in context.

The probability that it takes longer to packthan to box a system is the probability thatthe difference P - B is greater than zero.

What Can Go Wrong?

V Normal models: Weare told that both ran-dom variables follow Normal models.

V Independence: Wecan assume that thetimes it takes to pack and to box a systemare independent.

E(D) = E(P - B)=E(P) - E(B)

= 9 - 6 = 3 minutesSince the times are independent,Var(D) = Var(P - B)

= Var(p) + Var(B)= 1.52 + 12

Var(D) = 3.25SD(D) = V3.25 = 1.80 minutes

I'll model D with N (3, 1.80).

3

0-3z=--= -1.67

1.80

P(D> 0) = P(z> -1.67) = 0.9525

o

About 95% of all the stereo systems will requiremore time for packing than for boxing.

• Probability models are still just models. Models can be useful, but theyare not reality. Think about the assumptions behind your models. Are yourdice really perfectly fair? (They are probably pretty close.) But when youhear that the probability of a nuclear accident is 1/10,000,000 per year, is thatlikely to be a precise value? Question probabilities as you would data.

• If the model is wrong, so is everything else. Before you try to find themean or standard deviation of a random variable, check to make sure theprobability model is reasonable. As a start, the probabilities in your modelshould add up to 1. If not, you may have calculated a probability incorrectlyor left out a value of the random variable. For instance, in the insurance ex-ample, the description mentions only death and disability. Good health is by

Chapter 16 • Random Variables 379

far the most likely outcome, not to mention the best for both you and theinsurance company (who gets to keep your money). Don't overlook that.

• Don't assume everything's Normal. Just because a random variable iscontinuous or you happen to know a mean and standard deviation doesn'tmean that a Normal model will be useful. Youmust Think about whether theNormality Assumption is justified. Using a Normal model when it reallydoes not apply will lead to wrong answers and misleading conclusions.

To find the expected value of the sum or difference of random variables,we simply add or subtract means. Center is easy; spread is trickier. Watchout for some common traps.

• Watch out for variables that aren't independent. You can add expectedvalues of any two random variables, but you can only add variances of inde-pendent random variables. Suppose a survey includes questions about thenumber of hours of sleep people get each night and also the number ofhours they are awake each day. From their answers, we find the mean andstandard deviation of hours asleep and hours awake. The expected totalmust be 24 hours; after all; people are either asleep or awake.i The meansstill add just fine. Since all the totals are exactly 24 hours, however, the stan-dard deviation of the total will be O.We can't add variances here because thenumber of hours you're awake depends on the number of hours you'reasleep. Be sure to check for independence before adding variances.

• Don't forget: Variances of independent random variables add. Stan-dard deviations don't.

• Don't forget: Variances of independent random variables add, evenwhen you're looking at the difference between them.

• Don't write independent instances of a random variable with notationthat looks like they are the same variables. Make sure you write each in-stance as a different random variable. Just because each random variable de-scribes a similar situation doesn't mean that each random outcome will bethe same. These are random variables, not the variables you saw in Algebra.Write x, + X2 + X3rather than X + X + X. (S)

CONNECTIONSWe've seen means, variances, and standard deviations of data. We know that they estimate parame-ters of models for these data. Now we're looking at the probability models directly. We have onlyparameters because there are no data to summarize.

It should be no surprise that expected values and standard deviations adjust to shifts and changesof units in the same way as the corresponding data summaries. The fact that we can add variancesof independent random quantities is fundamental and will explain why a number of statisticalmethods work the way they do.

2 Although some students do manage to attain a state of consciousness somewhere between sleepingand wakefulness during Statistics class.