***SECTION 7.1*** Discrete and Continuous Random Variables

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1 ***SECTION 7.*** Discrete and Continuous Random Variables Samle saces need not consist of numbers; tossing coins yields H s and T s. However, in statistics we are most often interested in numerical outcomes such as the of heads in the four tosses. We will then use shorthand notation. For examle, let be the number of heads, then if our outcome is HHTH, then we write. Notice that can be. We call a because its values vary when the coin tossing is reeated. A is a variable whose value is a numerical outcome of a random henomenon. Usually denoted with caital letters near the end of the alhabet (, Y, etc.) Random variables are the basic units of distributions, which, in turn, are the foundations for inference. There are two tyes of random variables to be studied: and. Discrete Random Variables A has a countable number of ossible values. The of a discrete random variable lists the values and their robabilities: Value of : x x x x k Probability: k The robabilities i must satisfy two requirements:. Every robability i is a number between.. The sum of the robabilities is : k =. Find the robability of any event by adding the robabilities i of the articular values x i that make u the event. Examle : Automobile Defects A consumer organization that evaluates new automobiles customarily reorts the number of major defects on each car examined. Let x denote the number of major defects on a randomly selected car of a certain tye. A large number of automobiles were evaluated, and a robability distribution consistent with these observations is: Interret () x (x) Find and interret the robability that the number of defects is between and 5 inclusive

2 A robability histogram is a ictorial reresentation of a discrete robability distribution. Below is a robability histogram for the revious examle: Note: ) The height of each bar shows the robability of the outcome as its base ) Since the heights are robabilities, they add to ) All bars in a histogram are the same width 4) Histograms make it easy to quickly distributions Examle : Hot Tub Models Suose that each of four randomly selected customers urchasing a hot tub at a certain store chooses either an electric (E) or a gas (G) model. Assume that these customers make their choices indeendently of one another and that 40% of all customers select an electric model. This imlies that for any articular one of the four customers, P(E) = and P(G) = One ossible exerimental outcome is EGGE, where the first and fourth customers select electric models and the other two choose gas models. Because the customers make their choices indeendently, the multilication rule for indeendent events imlies that P(EGGE) = P( st chooses E and nd chooses G and rd chooses G and 4 th chooses E) = P(E) P(G) P(G) P(E) = (.4) (.6) (.6) (.4) =.0576 What is the robability for the exerimental outcome GGGE? Let us now consider x = the number of electric hot tubs urchased by the four customers Then we can consider the outcomes and robabilities as follows.

3 Outcome Probability x Value Outcome Probability x Value GGGG.96 0 GEEG EGGG.0864 GEGE GEGG.0864 GGEE.0576 GGEG GEEE GGGE.0864 EGEE EEGG EEGE EGEG.0576 EEEG.084 EGGE.0576 EEEE Notice that there are four different outcomes for which x =, so () results from summing the four corresonding robabilities: () = P( x = ) = P(EGGG) + P(GEGG) + P(GGEG) + P(GGGE) Similarly, () =.456 () =.56 (4) =.056 We can then summarize the robability distribution of x: The robability histogram for this distribution is: Find and interret the robability that at least two of the four customers choose electric models

4 Continuous Random Variables When we use the table of random digits to select a digit between 0 and 9, the result is a random variable. The robability model assigns robability to each of the 0 ossible outcomes. Suose that we want to choose a number at random between 0 and, allowing number between 0 and as the outcome. The samle sace is now an entire interval of numbers: How can we assign robabilities? We would like all outcomes to be, but we cannot assign robabilities to each individual value of x and then sum because there are many ossible values. So we will use a new way of assigning robabilities directly to events; that is, using. Any density curve has area exactly underneath it, corresonding to total robability. Examle : Alication Processing Time Define a continuous random variable x by x = amount of time (in minutes) taken by a clerk to rocess a certain tye of alication form. Suose that x has a robability distribution with density function.5 4 < x < 6 f ( x) = 0 otherwise The grah of f(x), the density curve, is shown below in (a). When the density curve is constant over an interval (resulting in a horizontal density curve), the robability distribution is called a distribution. It is esecially easy to use this density curve to calculate robabilities, because it just requires finding the area of rectangles using the formula area = ( base)( height ) The curve has ositive height, 0.5, only between x = 4 and x = 6. The total area under the curve is: b) P( 4.5 x 5.5) c) P( 5.5 x) < < = < = Interretation: 4

5 C A random variable takes all values in an of numbers. The robability distribution of is described by a density curve. The robability of any event is the area the density curve and the values of that make u the event. The robability model for a continuous random variable assigns robabilities to intervals of outcomes rather than to individual outcomes. In fact, all continuous robability distributions assign robability 0 to every individual outcome! Only intervals of values have ositive robability. This is clear when you think of finding area. * We can ignore the distinction between > and when finding robabilities for CONTINUOUS (but not discrete) random variables. Refer to examle (discrete): x 0 4 (x) P ( > ) = P ( ) = Normal Distributions as Probability Distributions The density curves that are most familiar to us are the Normal curves (from Chater ). Because any density curve describes an assignment of robabilities, Normal distributions are robability distributions. Examle 4: Internet for Information According to a recent Associated Press oll, aroximately 40% of American adults indicated they used the internet to get news and information about olitical candidates. Suose 40% of all American adults use this method to get their olitical information. What would haen if you randomly samled a grou of American adults and asked them if they used the internet to get this information. Define to be the % of your samle that would resond that the internet was their rimary source. Use the fact that the distribution of is aroximately N(0.4, 0.065) to answer the following questions. a) P( 0.4) b) P( 0.8 < < 0.4) c) P ( = 4.) 5

6 ***SECTION 7.*** Means and Variances of Random Variables In revious chaters, we moved from grahs to numerical measures such as and. Now we will make the same move to exand our descritions of the distributions of random variables. The Mean of a Random Variable The mean x of a set of observations is their ordinary average. The mean of a discrete random variable is also an of the ossible values of, but with an essential change to take into account the fact that outcomes need to be equally likely. The common symbol for the of a robability distribution is µ, the Greek letter mu. We used µ in Chater for the mean of a Normal distribution, so this is not a new notation. We will often be interested in random variables, each having a robability distribution with a mean. To remind ourselves that we are talking about the mean of we often write µ rather than simly µ. You will often find the mean of a random variable called the. Mean of a Discrete Random Variable Suose that is a discrete random variable whose distribution is Value of : x x x x k Probability: k To find the mean of, multily each ossible value by its robability, then add all the roducts: µ = x + x x k k = x i i Examle 5: Exam Attemts Individuals alying for a certain license are allowed u to four attemts to ass the licensing exam. Let x denote the number of attemts made by a randomly selected alicant. The robability distribution of x is as follows: x 4 (x)....4 Then x has mean value: 6

7 The Variance of a Random Variable The variance and the standard deviation are the measures of that accomany the choice of the mean to measure center. Just as for the mean, we need a symbol to distinguish the variance of a random variable from the variance s of a data set. We write the variance of a random variable as σ. The definition of the variance σ of a random variable is similar to the definition of the samle variance Variance of a Discrete Random Variable Suose that is a discrete random variable whose distribution is and that µ is the mean of. The variance of is s given in Chater. Here is the definition. Value of : x x x x k Probability: k ( ) ( )...( ) ( x ) = x + x + xk k σ µ µ µ = µ i i The standard deviation σ of is the square root of the variance. Examle 6: Defective Comonents A television manufacturer receives certain comonents in lots of four from two different suliers. Let x and y denote the number of defective comonents in randomly selected lots from the first and second suliers, resectively. The robability distributions for x and y are as follows: x 0 4 y 0 4 (x) (y) Probability histograms are given below: What can we say by examining the histograms? Find the variance and standard deviation for x and y. What can we conclude? 7

8 Statistical Estimation and the Law of Large Numbers To estimate µ, we often choose an and use the samle mean x to estimate the unknown oulation mean µ. Statistics obtained from robability samles are random variables because their values would in samling. The of statistics are just the robability distributions of these random variables. We will study samling distributions in Chater 9. It seems reasonable to use x to µ. An SRS should fairly reresent the oulation, so the mean x of the samle should be somewhere near the mean µ of the oulation. We don t exect x to be µ. We realize that if we choose another SRS, the luck of the draw will robably roduce a x. However, if we kee adding observations to our random samle, the statistic x is to get as close as we wish to the arameter µ and then stay that close. This remarkable fact is called the law of large numbers and it holds for oulation. Law of Large Numbers Draw observations at random from any oulation with finite mean µ. Decide how accurately you would like to estimate µ. As the number of observations drawn, the mean x of the observed values eventually the mean µ of the oulation as closely as you secified and then stays that close. Notice that as we increase the size of our samle, the samle mean x aroaches the mean µ of the oulation: Thinking about the Law of Large Numbers The gamblers in a casino may win or lose, but the casino will win because the law of large numbers says what the average outcome of many thousands of bets will be. An insurance comany deciding how much to charge for life insurance and a fast-food restaurant deciding how many beef atties to reare also rely on the fact that averaging over individuals roduces a result. 8

9 How large is a large number? The law of large numbers doesn t say how many trials are needed to guarantee a mean outcome close to µ. It all deends on the of the random outcomes. The more variable the outcomes, the more trials are needed to ensure that the mean outcome x is close to the distribution mean µ. Casinos understand this: the outcomes of games of chance are variable enough to hold the interest of gamblers. Only the casino lays often enough to rely on the law of large numbers. Gamblers get entertainment; the casino has a business. * Our intuition doesn t do a good job of distinguishing random behavior from systematic influences. This is also true when we look at data. We need statistical inference to sulement exloratory analysis of data because robability calculations can hel verify that what we see in the data is more than a random attern. Rules for Means Rule If is a random variable and a and b are fixed numbers, then µ a+ b= a+ bµ. Rule If and Y are random variables, then µ + Y= µ + µ Y. Examle 7: Linda Sells Cars and Trucks Linda is a sales associate at a large auto dealershi. She motivates herself by using robability estimates of her sales. For a sunny Saturday in Aril, she estimates her car sales as follows: Linda s estimate of her truck and SUV sales is: Cars sold: 0 Probability: Vehicles sold: 0 Probability: Take to be the number of cars Linda sells and Y the number of trucks and SUVs. The means of these random variables are: 9

10 Examle 7 continued At her commission rate of 5% of a gross rofit on ach vehicle she sells, Linda exects to earn $50 for each car sold and $400 for each truck or SUV sold. So her earnings are Thus, her mean earnings (best estimate of her earnings for the day) are Rules for Variances Two random variables and Y are if knowing that any event involving alone did or did not occur tells us about the occurrence of any event involving Y alone. Probability models often assume indeendence when the random variables describe outcomes that aear unrelated to each other. You should ask in each instance whether the assumtion of indeendence. Rules for Variances Rule If is a random variable and a and b are fixed numbers, then σa+ b= b σ. Rule If and Y are indeendent random variables, then σ = σ + σ + Y Y σ Y = σ + σy This is the addition rule for variances of indeendent random variables. As with data, we refer the of a random variable to the variance as a measure of. Examle 8: Proane Gas Consider the exeriment in which a customer of a roane gas comany is randomly selected. Suose that the standard deviation of the random variable x = number of gallons required to fill a customer s roane tank is known to be 4 gallons. The comany is considering two different ricing models: Model : $ er gal Model : service charge of $50 + $.80 er gal 0

11 Examle 8 continued The comany is interested in the variable y = amount billed For each of the two models, y can be exressed as a function of the random variable x: Model : ymodel Model : ymodel x 50.8 x Find the standard deviation of the billing amount variable. Examle 9: Luggage Weights A commuter airline flies small lanes between San Luis Obiso and San Francisco. For small lanes, the baggage weight is a concern, esecially on foggy mornings, because the weight of the lane has an effect on how quickly the lane can ascend. Suose that it is know that the variable x = weight of baggage checked by a randomly selected assenger has mean and standard deviation of 4 and 6, resectively. Consider a flight on which 0 assengers, all traveling alone, are flying. If we use xi to denote the baggage weight for assenger i (for i ranging from to 0), the total weight of checked baggage, y, is then y x x... x0 Note that y is a linear combination of the xi. Find the mean and standard deviation of y.

12 Combining Normal Random Variables So far, we have concentrated on finding rules for means and variances of random variables. If a random variable is Normally distributed, we can use its mean and variance to comute robabilities. What if we combine two Normal random variables? Any linear combination of indeendent Normal random variables is also Normally distributed. Examle 0: A Round of Golf Tom and George are laying in the club golf tournament. Their scores vary as they lay the course reeatedly. Tom s score has the N(0, 0) distribution, and George s score Y varies from round to round according to the N(00, 8) distribution. If they lay indeendently, what is the robability that Tom will score lower than George and thus do better in the tournament?

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