7.2 Day 2: Rules for Means and Variances

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? 1/2 1/3 1/6 CW NE EV R R R 1/2 2/3 3/4

Rules for Means Rule 1: If X is a random variable and a and b are fixed numbers, then μ a + bX = a + bμ X Rule 2: If X and Y are random variables, then μ X + Y = μ X + μ Y

Ex 1: Linda sells cars and trucks Find the mean of each of these random variables. μ x = 1.1 cars μ y = 0.7 trucks Cars Sold: 0123 Probability Trucks Sold:012 Probability

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 μ z = 350μ x + 400μ y = (350)(1.1) + (400)(0.7) = $650

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

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!

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. xixi pipi xipixipi (x i – μ x ) 2 p i μ x =σ x 2 =

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

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. μ X + Y = μ X + μ Y = $ $0.50 = $1.00 Find the variance of X + Y. σ 2 X + Y = σ X 2 + σ Y 2 = = Find the standard deviation of the total payoff. σ X + Y = Note that this is not the sum of the individual standard deviations ($ $15.80)

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