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Chapter 16 Random Variables In stats, what is a random variable? ► Denoted with a capital letter ► Value based on a random event ► Individual values denoted by the corresponding lowercase letter ► If all outcomes can be listed, we denote that as a discrete random variable ► If the amount of outcomes are infinite, we call it a continuous random variable ► The collection of all possible values and their probabilities are called the probability model Where do we see this outside of the classroom? ► Common use is in actuary science ► Actuary science is generally associated with insurance ► It is the science of estimating and evaluating risk Example 1: It’s a matter of life and death ► You are tasked with the job of setting prices for a life and disability insurance plan. Currently the payout for death is $10,000, a payoff for a person suffering a disability is $5000 and should a person have neither of the two happen the payout is $0 (Zero). The premium for the year is $50 a customer. In the past year 1/1000 people died, 1/500 people suffered a disability. The insurance company you work for wants to make a profit of over 100% on this policy. Will they need to set a new premium for the upcoming year? How to solve ► First in order to solve we need to understand what our goal is ► We need to find the average cost of the policy per person. ► This specific type of average is called an expected value ► Using the least common denominator we will estimate the average cost of the policy for 1000 people Computation ► Find the sum of the money being spent on 1000 policy holders using last years statistics ► 10,000*(1)+5,000*(2)+0*(997) ► =20,000 ► Second take that sum and divide it by the 1000 policy holders to find the average cost per policy holder ► 20,000/1,000 ► =20 ► Now put it in context ► If each policy holder spends $50 per policy and the expected cost to the insurance company is $20 per policy, the profit for the company is 150% (50/20=1.5). So based on last years death and disability statistics, there is no need to change the cost of the policy. ► ***When CUSSing, the expected value is considered the center. Spread Out ► If we can find a mean, we can find a standard deviation ► To find the standard deviation of a discrete variable, we must first find the deviation of each possibility. ► Then we square each deviation to find the variance. Since the variance is the expected value of the squared deviations we must multiply each squared deviation by the probability associated with it. ► Finally to find the standard deviation, we take the square root of the variance ► We will use our first example to help illustrate the process AP Speak ► In this problem, the variable was discrete ► The formula for calculating the expected of a discrete variable is as follows: Computation and AP style breakdown ► A)Define the random variable and construct the probability model Define the variable in a complete sentence Put the probability model in a chart ► B) What is the expected value of the repair Show all the computations that you have done and include the formula that you use ► C) What does this mean in context? Write a paragraph to make sense of what you have done and what it means. Make sure to stay away from the use of pronouns. Example 2: Extended Warranty? Do I really need it? ► So when I make a purchase I always research the product and price, but never the cost of an extended warranty. Too many times an extended warranty is offered by some pushy salesmen working on a commission and they often feel like a rip- off. It’s a gamble and the house almost always wins. With that being said, in the following problem help me decide if it is a better deal to buy the extended warranty. Car Problems ►A car mechanic identified two possibilities for the cause of a broken air conditioner. Either there is dirt in the control unit or the unit is broken down and needs to be replaced. 75% of the time, this problem can be fixed by simply cleaning the control unit for $60. In order to find out if the control unit is broken it first must be cleaned. The cost of replacing the unit is an additional $100 for the part and an additional $40 for labor. Before working on my air conditioner, the mechanic offers to replace the whole unit for $120. Am I better off just having the unit replaced? Computation ► Deviations Death: 10,000-20=9980 Disability: 5,000-20=4980 Neither: 0-20=-20 ► Variance 1 2 2 2 997 Var ( X ) 9980 2 4980 (20 ) 149 ,600 1000 1000 1000 ► Standard Deviation SD( X ) 149,600 $386.78 AP Speak ► The formulas you need to know Variance 2 Var( X ) ( x )2 P( X ) Standard Deviation SD( X ) Var( X ) Example 3: Computer Glitch ► So you own your own computer business and just made your first big sale. You were so excited that you shipped the 2 computers your client bought that same day. Unfortunately though, your regional manager got into the stock room again and mixed up the refurbished computers with the new computers. The 2 that you shipped were randomly selected and are well on their way. You had 15 new computers and 4 refurbished computers in your inventory. If your client receives 2 new computers you can breathe easy. If he gets one refurbished it will cost you $100 to ship and replace. If both are refurbished, your client will do business elsewhere costing you $1000. ► What is the expected value of your loss? ► What is the standard deviation of your loss? AP Practice ► Writeout a plan to ensure the reader you understand the question. I am going to find my company’s expected loss for incorrect shipping. ► To create an equation first define the random variable. X=amount loss R=refurbished N=New ► To help you figure out probabilities, draw a tree diagram. (You can not create your equation until you do this) Creating the model and computations ► Using the tree create the model Outcome X (Cost) P(X=x) 2 Refurbished 1000 P(RR)=.057 1 Refurbished 100 P(NR U RN)=.2095+.20 95=.419 0 Refurbished 0 P(NN)=.524 Computations ► To find the expected value multiply the cost of the error times the probability for each possibility. Then take the sum E(X) = 0(.524)+100(.419)+1000(.057) =$98.9 Variance and Standard Deviation ► Variance Var(X)= (0 98.90) 2 (.524) (100 98.90) 2 (.419) (0 98.90) 2 (.057) 51408.79 ► Standard Deviation = 51408 .78 $226 .735 So what does it all mean? ►I expect the mistake to cost $98.90, with a standard deviation of $226.74. There is such a large standard deviation because of the large gap in cost between 1 and 2 refurbished computers. I will be demoting my regional manager to assistant regional…no assistant to the regional manager. Day 1 Homework ► Page 381 – 385 4, 7, 8, 16, 18 Day 2 Problem of the Day ► Using today’s handout go through the process of think-show-tell for the following problem: A couple plans to have children until they get a girl, but they agree that they will not have any more than three children even if all 3 are boys. Assuming that gender probabilities are equivalent. ► A) Create a probability model for the number of children they will have ► B) Find the expected number of children ► C) Find the expected number of boys Day 2 ► When we first introduced mean and standard deviation we said that when we add a constant to E ( X c) E ( X ) c the mean, or shift the mean, the variance and Var ( X c) Var ( X ) standard deviation stayed the same. The same is true when we have an expected value What about if we use multiplication? ► When we multiplied the data points by a constant, the mean E (aX ) aE( X ) was multiplied by the constant and the Var(aX ) a Var( X ) 2 variance by the constant squared. The same holds true with expected value Things to think about ► Yesterday we simplified the process of insurance for the sake of showing you how to calculate the expected value, now lets investigate a little deeper. ► Does it cost the same to insure one person for at payouts of $20,000 and $10,000 as it does to insure two people at rates of $10,000 and $5,000? ► Or if you would rather think about it this way, is the risk the same? ► Does either sentence phrasing make a difference? ► What if that number changed to 1000 people? ► Let’s discuss…What I want to do here is get an idea of what would seem logical to you. Hopefully we came to a consensus that risks and payouts are not the same ► Main reason it is different is that the odds of one person dying are greater than 2 people both dying. ► The risk gets spread out among more people. ► As the risk spreads out the variance becomes more predictable. Day 3 ► Nothing New ► Nothing Due Take 2 People as an example. We will use our data from yesterday’s insurance question. ► Married Mac ► Single Mac Mac (X) CJ (Y) Payouts are doubled ► Are our risk factors since he is only one person and he is independent…generally insured for twice as speaking? much. Since they are then we Use E(2X) and Var(2X) will use E(X+Y) and Var(2X)=4(149600) Var(X+Y) =598,400 ► Using the data from Standard deviation = yesteday $773.56 compared to Var(X)=149,600 $546.99 for Married Var(Y)=149,60 Mac Var(X+Y)=299,200 Example 5: Intergalactic Currency Exchange ► You decide to take an intergalactic trip to Tatooine to explore the deserts. In order to do so you must trade in your old pod racer for a landspeeder. Currently your pod is worth 6,940 Galactic credits with a standard deviation of 250 Galactic credits. A land speeder is going for 65,000 Imperial Credits with a standard deviation of 500 Imperial Credits. 1 Galactic credit is worth 43 Imperial credits. How much money can you expect to have in your pocket to spend on the Cantina in Mos Eisley after your sale and purchase? Answers: ► Think: I want to estimate my spending money at Mos Eisley. ► My variables are G=sale of my pod racer I=price of a landspeeder M=Profits for Mos Eisley ► Equation M=43G – I ► Independence? From the empire no, for my purchases YES! Answers continued ► Show Expected Value E(M)=E(43G - I) =43E(G)-E(I) =43(6940)-65,000 =233,420 Imperial Credits Answers continued ► Show Variance and Standard Deviation Var(M)=Var(43G – I)***This is the proper notation since we are subtracting the two values however…. =Var(43G)+Var(I) ***We always add variances =43*43Var(G)+Var(I) =1849(250*250)+(500*500) Var(M) = 115,812,500 Standard deviation is the square root of the previous answer, 10,762 Imperial Credits Conclusion ►I can expect to have 233,420 Imperial credits with a standard deviation of 10,762 Imperial Credits ► This is a profit of 5428 Galactic Credits with a standard deviation of 250 Galactic credits Continuous Random Variable ►A variable that can take on any value. Time is a continuous random variable ► What type of models can occur Normal, skewed, symmetric, bimodal. depth will we go into for the AP ► What exam? Our main concern is when they are normal. We will use what we know about normal distributions to make calculations. What’s the difference between Continuous and Discrete Random variables? ► Values of discrete variables are all known and have a probability associated with them ► Continuous variables must fit in some kind of model as listed above but can be anywhere on the spectrum. (i.e. the possible outcomes are infinite.) ► When a continuous variable is normal, the process of calculating the expected value, variance, and standard deviation are the same as in a discrete variable. Example 4: Baseball’s been very very good to me ► While working for the flyers, it took an average of 100 seconds to serve a customer with a standard deviation of 50 seconds. (What can I say occasionally a fried carrot worked its way into the hot dog bun). If you came to my stand with two people in front of you how long would you expect to wait? What is the standard deviation of your wait time? What assumptions must you make about those in front of you? Answers ► How long do you expect to wait? E(customer 1 + customer 2)=100+100 200 seconds or 3 minutes and 20 seconds ► What is the standard deviation of your wait time? Variance is the standard deviation squared. Since we are adding the two, we must first convert them back to variances to solve. Variance =50 squared + 50 squared = 5000 The square root of that is 70.7 seconds ► What assumptions must you make? That each customer ahead of you in line is independent from the other. Example 6: Gone are the days of the boom box ► Back in my day it was cool to carry around a boom box. You were something with your big ole box and your Public Enemy. I decide I want to bring them back into style and hire you all to work in the packing department. There are two parts to the packing process, preparing the boom boxes for shipping and boxing them. As a class, the time it takes you to prepare the packages (Stage 1) is normally distributed with a mean of 9 minutes standard deviation of 1.5 minutes. The time it takes to put the boom boxes into their shipping box (Stage 2) is also normally distributed with a mean of 6 minutes and a standard deviation of 1 minute Questions ► What is the probability that packing two consecutive systems takes over 20 minutes? ► What percentage of stereos take longer to pack than to box? Where do we begin? ► Describe what you are doing Rephrase the question ►Iwant to estimate the probability of packing two consecutive boom boxes takes over 20 minutes Define the variables ► P1 = Packing first boom box ► P2= Packing Second Boom Box ► T = Total packing time Create an equation ►T = P1 + P2 ► Do I have to do this? How many points will you take off if I don’t? YES! And Most to all points for this part will be lost if you chose not to Showing the Answer & Conclusion Part 1 ► E(P1+P2) = 18 minutes ► Var (T) = 4.5 ► SD(T) = 2.12 ► Show a Normal Curve See board ► Calculate a z score for 20 minutes ► Z score = .94 ► Probability associated with that score on the right tail is 17.36% ► We can say that there is a 17.36% chance that the it will take us more than 20 minutes to pack 2 boom boxes. Part 2 Thinking out the plan ► Describe what you are doing Rephrase the question ►I would like to estimate how often the boom boxes will take longer to pack than to box. Define the variables ►P = Packing the boom box ► B = boxing the Boom Box ► D = difference in times to pack and box a system Create an equation ►D =P–B Ultimately what we are looking for is for P – B to be greater than zero, that would mean that packing would take longer than boxing. Day 4 January 22, 2010 Pick up Practice Quiz C Due today Problems 28, 29 Day 4 Problem of the day ► Find the expected value, and the standard deviation of the following data. The top row is x and the bottom row is P(X) 100 200 300 400 .1 .2 .5 .2 Are the conditions met? ► Is this a normal distribution? The text of the problem stated that each distribution was normal. ► Is there independence? The packing of the boom box and the boxing of the boom box should not effect each other. ***Yes, I know that the quality of the packing would have an effect on the boxing, but we are assuming that the boom box does not leave stage 1 unless it is in perfect condition. Show and Tell ► First let’s find the expected value E(D) = E(P – B) =E(P) – E(P) 9 – 6 = 3 minutes ► Find the variance and standard deviation Var(D) = Var(P – B) Var(P) + Var(B) 1.5*1.5 + 1*1 Var(D) = 3.25 SD (D) = the square root of 3.25 which is approximately 1.8 Show a picture ► Findthe z score associate with the difference (0-3)/1.8 = -1.67 ► Seethe board for the picture ► P(D>0)=P(z>-1.67)=.9525 ► Conclusion About 95% of boom boxes will take more time in stage 1 than in stage 2 Day 5 Announcements ► Homework check Quiz C ► Review Today Problem Solving Recap Expected Value Game ► Quiz Tomorrow on Chapter 16 Step By Step Recap ► Think Plan: Actually state what you are going to try and solve ► In order to decide if I need to change the cost of the life insurance policy I must… Variable: Define the discrete variable in words ►X is the cost of the insurance policy on the insurance company Plot ► Make a picture of the probabilities. Use a tree diagram to illustrate this Model ► Createa table of all possible values and outcomes of the random variable Step By Step Recap Continued ► Show Find the expected value ►Show formulas and computations Find the Variance ►Show formulas and computations Find the Standard Deviation ►Show formulas and computations Step By Step Recap Continued ► Tell Conclusion ►Write out a paragraph in complete sentences ►Use the context of the problem and avoid pronouns ►Discuss how you found your answers and interpret what they mean