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Combining and “Adjusting” Discrete Random Variables Section 7-2 Two Rules regarding means Rule 1 for Means: If you’re doing a linear transformation, play it just as we did before: New = a + b Old …where ◦ a is a constant you might add to every value of the random variable, and ◦ b is a constant by which you might multiply every value of the random variable Example: Gain Electronics (page 419) X 1000 3000 5000 10,000 Probability .1 .3 .4 .2 X represents numbers of radios for military applications. The probabilities are the estimates by the company’s executives. For instance, the executives think there’s a .1 probability that they’ll only sell 1000 units, a .3 probability of selling 3000 units, etc. Question: What’s the best guess of the number of units that will be sold next year for military applications? Example: Gain Electronics (page 419) X 1000 3000 5000 10,000 Probability .1 .3 .4 .2 Question: What’s the best guess of the number of units that will be sold next year? (Hint: “Best guess” = “Expectation” = “Expected Value” = Mean) E(x) = μx = _______________ Example: Gain Electronics (page 419) X 1000 3000 5000 10,000 Probability .1 .3 .4 .2 Question: If we profit $2000 for every radio, what will be our expected total profit? profit = 2000(radios) = 2000(5000) = $10,000,000 Notice: a = 0; b = 2000. In other words, there really is an a and b. a simply happens to equal 0. Also notice that you would get this same answer if you did indeed multiply each value of X by 2000, then recalculate the mean. Rule 2 for Means: If you’re combining two variables through addition (or subtraction), the mean of the new distribution is the sum (or diff.) of the two distributions separately. μX+Y = μX + μY Example: Gain Electronics (page 419) Y 300 500 750 P(Y) .4 .5 .1 New random variable Y represents numbers of radios for civilian applications. Question: What’s the best guess of the total number of units that will be sold next year? (Military and Civilian?) μX+Y μMil + Civ = μX + μY = μMil + μCiv = 5000 + 445 = 5445 radios Again—”Best Guess” = “Expected Value” = “Mean” Throw it all together… What’s the expected total income for Gain Electronics? Use both rules. E(total income) = E(military income) + E(civilian income)\ E(total income) = $2000 (5000 radios) + $3500 (445 radios) = $11, 557, 500 ◦ Notice—There’s multiplying and adding going on! Rules for Variance Looky here! new b old ◦ Remember how when you’re performing a linear transformation, Standard Deviation is only effected by multiplicative constants? (Chapter 1, p. 55) ◦ Well—what happens when you square both sides? 2 (b old ) b 2 new 2 2 old Rules for Variance 1. To find the standard deviation of a combined random variable, add variances, then take the square root. NEVER EVER ADD STANDARD DEVIATIONS! 2. Even when combining random variables through subtraction, add variances. Rules for Variance ◦ Again ◦ Example: The distribution of random variable X is N(24, 4). Find the Var(X) if you triple every value. 2 (b old ) b 2 new 2 new b 2 2old 2 new 32 42 96 2 2 old In your textbook—Go to p. 428 Do problem 7.46 What about combining more than two Random Variables? Mean: Add all the means together. Standard Deviation: Add all the variances together, then take the square root of the sum. Example: Team of 4 students Teams of four students are randomly selected from a class whose grade distribution is in the table. 4 3 2 1 .2 .4 .2 .2 Find the total mean and standard deviation for combinations of four students Example: Teams of 4 students Mean: Take mean for the singlestudent distribution (2.6), and add it four times. µ = 10.4 4 3 2 1 .2 .4 .2 .2 Standard Deviation: Add the variance four times, then take the square root. 1.04 1.04 1.04 1.04 2.04