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Lesson Objectives At the end of the lesson, students can: โข Perform linear transformations on Random Variables. โข Determine the mean and standard deviation of transformed Random Variables. โข Determine the mean and standard deviation of sums and differences of Random Variables. Linear Transformation Review In Unit 1, we studied the effects of transformations on the shape, center and spread of a distribution of data. ๐๐๐๐ = ๐ + ๐๐๐๐๐ Adding (or subtracting) a constant: Adding the same number, ๐, to each observation โข Adds ๐ to measures of center โข Does not change shape or measures of spread Multiplying (or dividing) each observation by constant, ๐ : โข Multiplies (divides) measures of center by ๐ โข Multiplies (divides) measures of spread โข Does not change shape of distribution Linear Transformation of Random Variables How are the probability distributions of random variables affected by similar transformations? Letโs follow an example through thisโฆ Peteโs Jeep Tours Peteโs Jeep Tours offers a popular half-day trip in a tourist area. There must be at least 2 passengers for the trip to run, an the vehicle will hold up to 6 passengers. Hereโs the probability distribution: Number of Passengers (X) Probability 2 .15 3 .25 4 .35 5 .20 Draw the probability histogram. What is the mean, ๐๐ ? What is the standard deviation, ๐๐ , and variance, ๐๐ 2 ? 6 .05 Peteโs Jeep Tours Peteโs charges $150 per passenger. Let C = the amount of money Pete collects on a randomly selected trip. Show the probability distribution of C: C = 150 * X Total collected (C) Probability Draw the probability histogram. What is the mean, ๐๐ถ ? What is the standard deviation, ๐๐ถ , and variance, ๐๐ถ 2 ? How do these relate to the numerical summaries for X? (shape, center, spread) Peteโs Jeep Tours It costs Pete $100 to buy permits, gas, and a ferry pass for each half-day trip. The amount of profit that Pete makes from the trip, V, is the total amount of money, C, minus $100. What is the probability distribution of V. V = C - 100 Total collected (V) Probability Draw the probability histogram. What is the mean, ๐๐ ? What is the standard deviation, ๐๐ , and variance, ๐๐ 2 ? How do these relate to the numerical summaries for C? for X? (shape, center, spread) Rules for MEANS RULES FOR MEANS Means behave like averages! Rule 1: If X is a random variable and a and b are fixed numbers, then ฮผa+bx = a + bฮผx ฮผ aโbx = a โ bฮผx Rule 2: If X and Y are random variables, then ฮผ x + y = ฮผx + ฮผy ฮผ x โ y = ฮผx โ ฮผy In other words, the mean of the sum = sum of the means and the mean of the difference = difference of the means. Linear Transforms of Random Variables EXAMPLE: Consider the data on the distribution of the number of communications units sold by the military division and the civilian division. Military Division: # of Units Sold (X) Probability 1000 .1 3000 .3 300 .4 500 .5 5000 10,000 .4 .2 Civilian Division: # of Units Sold (Y) Probability 750 .1 (a) Find the mean # of units sold collectively by both divisions. Let X = Let Y = ฮผx = ฮผy = ฮผX+Y = Linear Transforms of Random Variables EXAMPLE: Consider the data on the distribution of the number of communications units sold by the military division and the civilian division. (b) The company makes a profit of $2,000 per military units sold and $3500 on each civilian unit sold. What will next yearโs mean profit from military sales be? What will next yearโs mean profit from civilian sales be? (c) Suppose we multiplied each value of X by 2 and added 10. (2X + 10). How would this affect the mean? Rules for Variances RULES FOR VARIANCE Rule 1: If X is a random variable and a and b are fixed numbers, then ฯ2a+bx = b2ฯ2x ฯ2a-bx = b2ฯ2x NOTE: Multiplying X by a constant โbโ multiplies the variance of X by the โb2โ. The variance of X + a is the same as the variance of X. Rules for Variance Rule 2: If X and Y are independent random variables, then ฯ2x+y = ฯ2x + ฯ2y ฯ2x-y = ฯ2x + ฯ2y s.dev. = ฯ2x+y = (ฯ2x + ฯ2y ) In other words: Add the variances, then take square root (This is called the โAddition rule for variances of independent random variables.โ) Peteโs Sister Erin Peteโs sister, Erin, who lives in another part of the country, decided to join the business. She has a slightly smaller vehicle. The probability distribution for her business is: Number of Passengers (Y) Probability 2 .3 3 .4 4 .2 5 .1 Draw the probability histogram. What is the mean, ๐๐ ? What is the standard deviation, ๐๐ , and variance, ๐๐ 2 ? What is the average number of passengers Pete and Erin expect to have on their tours on a randomly selected day? Independent Random Variables If knowing whether any event involving โX alone has occurredโ tells us nothing about the occurrence of any event involving โY aloneโ, and vice versa, then X and Y are independent random variables. Are X, the number of Peteโs passengers on a random day, and Y, the number of Erinโs passengers on a random day, independent? Peteโs and Erinโs Combined Business Pete and Erin looked at their businesses together, where T = X + Y. Assuming their businesses are independent, the probability distribution for their combined businesses is: Number of Passengers (T) Probability 4 5 6 .045 .135 .235 7 .265 8 9 .190 .095 10 11 .030 .005 What is the mean, ๐ ๐ ? What is the standard deviation, ๐๐ , and variance, ๐๐ 2 ? NOTE: You have the know-how to compute the probabilities for the above table yourself! (See page 366 in book) Rules for Means and Variances Rules for Variance NOTE: When random variables are not independent, the variance of their sum depends on the relationship between them as well as on their individual variances. We use ฯ (Greek letter โrhoโ) for the correlation between two random variables. The correlation ฯ is a number between -1 and 1 that measures the strength and direction of the linear relationship between the two variables. The correlation between two independent random variables is zero. We will not be looking into detail with variance of not independent random variables. EXAMPLE: Earlier, we defined X = the number of passengers on Peteโs trip, Y = the number of passengers on Erinโs trip, and C = the amount of money that Pete collects on a randomly selected day. We also found the means and standard deviations of these variables: ฮผx = 3.75 ฮผy = 3.10 ฮผC = 562.50 ฯx = 1.090 ฯy = 0.943 ฯC = 163.50 (a) Erin charges $175 per passenger for her trip. Let G = the amount of money that she collects on a randomly selected day. Find the mean and standard deviation of G. EXAMPLE: Earlier, we defined X = the number of passengers on Peteโs trip, Y = the number of passengers on Erinโs trip, and C = the amount of money that Pete collects on a randomly selected day. We also found the means and standard deviations of these variables: ฮผx = 3.75 ฮผy = 3.10 ฮผC = 562.50 ฯx = 1.090 ฯy = 0.943 ฯC = 163.50 (b) Calculate the mean and the standard deviation of the total amount that Pete and Erin collect on a randomly chosen day. Linear Transformations of Random Variables EXAMPLE: A large auto dealership keeps track of sales made during each hour of the day. Let X= the number of cars sold during the first hour of business on a randomly selected Friday. The probability distribution of X is: # of Cars Sold (X) Probability 0 .3 1 .4 2 .2 3 .1 The mean is ๐๐ = 1.1 ; the standard deviation is ๐๐ = 0.943 . Suppose the dealershipโs manager receives a $500 bonus for each car sold. Let Y = the bonus received. Find ๐๐ and ๐๐ . ๐๐ = (500)(1.1) = $550 ๐๐ = (500)(0.943) = $471.50 Linear Transformations of Random Variables EXAMPLE: A large auto dealership keeps track of sales made during each hour of the day. Let X= the number of cars sold during the first hour of business on a randomly selected Friday. The probability distribution of X is: # of Cars Sold (X) Probability 0 .3 1 .4 2 .2 3 .1 The mean is ๐๐ = 1.1 ; the standard deviation is ๐๐ = 0.943 . To encourage customers to buy cars, the manager spends $75 to provide coffee and doughnuts. The managerโs net profit, T, on a random Friday is the bonus earned minus $75. Find ๐ ๐ and ๐๐ . ๐ ๐ = (500)(1.1) โ 75 = $475; ๐๐ = (500)(0.943) = $471.50 Combining Random Variables Example: A large auto dealership keeps track of sales and lease agreements made during each hour of the day. Let X = # of cars sold, and Y = # cars leased during the first hour of business on a randomly selected Friday. Based on previous records, the probability distributions of X and Y are as follows: (continued on next slide . . .) Example continued. . . Cars Sold, xi 0 1 2 3 Probability, pi 0.3 0.4 0.2 0.1 Cars Leased, yi 0 1 2 Probability, pi 0.4 0.5 0.1 Example continued. . . µx = 1.1; ฯx = 0.943 and µY = 0.7; ฯY = 0.64 Define T = X + Y. 1) Find and interpret µT. µT = µx + µY = 1.1 + 0.7 = 1.8. On average, this dealership sells or leases 1.8 cars in the first hour of business on Fridays. 2) Compute ฯT assuming that X and Y are independent. ฯT = (0.943)2 +(0.64)2 = 1.14 Example continued . . . 3) The dealershipโs manager receives a $500 bonus for each car sold and a $300 bonus for each car leased. Find the mean and standard deviation of the managerโs total bonus, B. Show work! µB = 500(1.1) + 300(0.7) = $760 ฯB = ๐๐๐ ๐ (๐. ๐๐๐)๐ +(๐๐๐)๐ (๐. ๐๐)๐ = $๐๐๐. ๐๐ Combining Normal Random Variables If a random variable is Normally distributed, we can use its mean and variance to compute probabilities. What if we combine two normal random variables? Any linear combination of independent Normal random variables is also Normally distributed! If X and Y are independent Normal random variables and a and b are any fixed numbers, then aX + bY is also Normally distributed. You must communicate this fact about the distributionโs shape! Combining Normal Random Variables EXAMPLE: Tom and George are avid golf players. Their scores vary as they play the course repeatedly according to the following distributions: Tomโs score X: N(110, 10) Georgeโs score Y: N(100, 8) If they play independently, what is the probability that Tom will score lower than George (and thus do better in the tournament)? APPLES! Suppose that a certain variety of apples have weights that are approximately Normally distributed with a mean of 9 oz and a standard deviation of 1.5 oz. If bags of apples are filled by randomly selecting 12 apples, what is the probability that the sum of the weights of the 12 apples is less than 100 oz? P(X<100) = 0.0620 Speed Dating โข To save time and money, many single people have decided to try speed dating. At a speed-dating event, women sit in a circle, and each man spends about 5 minutes getting to know a woman before moving on to the next one. โข Suppose that the height M of male speed daters follows a Normal distribution, with a mean of 70โ and a standard deviation of 3.5โ, and suppose that the height F of female speed daters follows a Normal distribution, with a mean of 65โ and a standard deviation of 3โ. โข What is the probability that the man is taller than the woman in a randomly selected speed-dating couple? Use 4-step process! โข P(M>F) = P(D>0), where D = M-F. P(D>0) = 0.8610 Lesson Objectives At the end of the lesson, students can: โข Perform linear transformations on Random Variables. โข Determine the mean and standard deviation of transformed Random Variables. โข Determine the mean and standard deviation of sums and differences of Random Variables.