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7.2 Day 2: Rules for Means and Variances

This text explains the rules for calculating means and variances and applies them to various scenarios. It also discusses the concept of probability and its application in determining the likelihood of certain events.

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7.2 Day 2: Rules for Means and Variances

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  1. 7.2 Day 2: Rules for Means and Variances

  2. 3/4 R CW 1/2 R 2/3 1/3 NE 1/2 1/6 R EV Probability WARM UP A travel agent books passages on three different tours, with half of her customers choosing Caribbean Waters (CW), one-third choosing New England’s Historic Trail (NE), the rest choosing European Vacation (EV). The agent has noted that three-quarters of those who take CW return to book passage again, two-thirds of those who take NE return and one-half of those who take EV return. If a customer does return, what is the probability that the person went on NE?

  3. Rules for Means Rule 1: If X is a random variable and a and b are fixed numbers, then μa+ bX = a + bμX Rule 2: If X and Y are random variables, then μX + Y = μX + μY

  4. Ex 1: Linda sells cars and trucks • Find the mean of each of these random variables. μx = 1.1 cars μy = 0.7 trucks

  5. At her commision rate of 25%, Linda expects to earn $350 for each car sold and $400 for each truck sold. So her earnings are Z = 350X + 400Y • Combining rules 1 and 2, her mean earnings are μz = 350μx + 400μy = (350)(1.1) + (400)(0.7) = $650

  6. Rules for Variances Adding a constant a to a random variable changes its mean but does not change its variability. Rule 1: If X is a random variable and a and b are fixed numbers, then σ2a + bX = b2σx2 Rule 2: If X and Y are independent random variables, then σ2X+Y = σx2 + σY2 σ2X – Y = σx2 + σY2 This is the addition rule for variances of independent random variables.

  7. Rules for Standard Deviations • Standard deviations are most easily combined by using the rules for variances rather than by giving separate rules for standard deviations. • Note that variances of independent variables add, standard deviations do not generally add!

  8. Ex 2: Winning the lottery Recall that the payoff X of a Tri-State lottery ticket is $500 with probability 1/1000 and $0 the rest of the time. a) Calculate the mean and variance. 0 0.24975 0.5 249.50025 0.5 249.75

  9. Games of chance typically have large standard deviations. Large variability makes gambling more exciting! b) Find the standard deviation. σx = $15.80 c) If it cost $1 to buy a ticket, what is the mean amount that you win? μw = μx – 1 = -$0.50 Note that the variance and standard deviation stay the same by Rule 1 for Variances. Basically, you lose an average of 50 cents on a ticket.

  10. d) Suppose that you buy two tickets on two different days. These tickets are independent due to the fact that drawings are held each day. • Find the mean total payoff X + Y. μX+ Y = μX + μY = $0.50 + $0.50 = $1.00 • Find the variance of X + Y. σ2X + Y = σX2 + σY2 = 249.75 + 249.75 = 499.50 • Find the standard deviation of the total payoff. σX + Y = Note that this is not the sum of the individual standard deviations ($15.80 + $15.80)

  11. Ex 3: SAT Scores It wouldn’t make sense to add the standard deviations, due to the fact that the test scores are not independent. • Below are the means and standard deviations of SAT scores at a certain college. SAT Math Score X μx = 519 σx = 115 SAT Verbal Score Y μY = 507 σY = 111 • Find the mean overall SAT Score. μX + Y = μX + μY = 519 + 507 = 1026

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