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Warm Up • How do I know this is a probability distribution? • What is the probability that Mary hits x=# red lights p(x) 0 0.05 1 0.25 2 0.35 3 0.15 4 0.15 5 0.05 exactly 3 red lights? • What is the probability that she gets at least 4 red lights? • What is the probability that she gets less than two? • Find the mean & standard deviation. Find Mean & Standard Deviation: x=# books read P(x) 0 0.13 1 0.21 2 0.28 3 0.31 4 0.07 Ex. 1. 2. 3. x = possible winnings P(x) 5 0.1 7 0.31 8 0.24 10 0.16 14 0.19 Find the mean Find the Standard Deviation Find the probability that x is within one deviation from the mean. 500 raffle tickets are sold at $2 each. You bought 5 tickets. What’s your expected winning if the prize is a $200 tv.? There are four envelopes in a box. One envelope contains a $1 bill, one contains a $5, one contains a $10, and one a $50 bill. A person selects an envelope. Find the expected value of the draw. What should we charge for the game for it to be fair? A person selects a card from a deck. If it is a red card, he wins $1. If it is a black card between or including 2 and 10, he wins $5. If it is a black face card, he wins $10, and if it is a black ace, he wins $100. Find the expectation of the game. What would it be if it cost $10 to play? What should I charge to make it a fair game? On a roulette wheel, there are 38 slots numbered 1 through 36 plus 0 and 00. Half of the slots from 1 to 36 are red; the other half are black. Both the 0 and 00 slots are green. Suppose that a player places a simple $1 bet on red. If the ball lands in a red slot, the player gets the original dollar back, plus an additional dollar for winning the bet. If the ball lands in a different-colored slot, the player loses the dollar bet to the casino. What is the player’s average gain? LINEAR TRANSFORMATIONS Section 6.2A Remember – effects of Linear Transformations • Adding or Subtracting a Constant • Adds “a” to measures of center and location • Does not change shape or measures of spread • Multiplying or Dividing by a Constant • Multiplies or divides measures of center and location by “b” • Multiplies or divides measures of spread by |b| • Does not change shape of distribution Adding/Subtracting a constant from data shifts the mean but doesn’t change the variance or standard deviation. • E X c E(X ) c • Var X c Var ( X ) Multiplying/Dividing by a constant multiplies the mean and the standard deviation. E ( aX ) aE ( X ) aX = a ∙ Var ( aX ) a Var ( X ) 2 Pete’s Jeep Tours offers a popular half-day trip in a tourist area. The vehicle will hold up to 6 passengers. The number of passengers X on a randomly selected day has the following probability distribution. He charges $150 per passenger. How much on average does Pete earn from the half-day trip? # Passengers Prob 2 0.15 3 0.25 4 0.35 5 0.2 6 0.05 Pete’s Jeep Tours offers a popular half-day trip in a tourist area. The vehicle will hold up to 6 passengers. The number of passengers X on a randomly selected day has the following probability distribution. He charges $150 per passenger. What is the typical deviation in the amount that Pete makes? # Passengers Prob 2 0.15 3 0.25 4 0.35 5 0.2 6 0.05 What if it costs Pete $100 to buy permits, gas, and a ferry pass for each half-day trip. The amount of profit V that Pete makes from the trip is the total amount of money C that he collects from the passengers minus $100. That is V = C – 100. So, what is the average profit that Pete makes? What is the standard deviation in profits? 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. Based on previous records, the probability distribution of X is shown below. Suppose the dealership’s manager receives a $500 bonus from the company for each car sold. What is the mean and standard deviation of the amount that the manager earns on average? # cars sold Prob 0 0.3 1 0.4 2 0.2 3 0.1 Suppose the dealership’s manager receives a $500 bonus from the company for each car sold. To encourage customers to buy cars on Friday mornings, the manager spends $75 to provide coffee and doughnuts. Find the mean and standard deviation of the profit the manager makes. # cars sold Prob 0 0.3 1 0.4 2 0.2 3 0.1 Variance of y = a + bx • Relates to slope. y b x b 2 y 2 x 2 y b x 2 2 var( y ) b 2 y b 2 2 x 2 2 var( x ) Effects of Linear Transformation on the Mean and Standard Deviation if = + . = + = *Shape remains the same. Example: Three different roads feed into a freeway entrance. The number of cars coming from each road onto the freeway is a random variable with mean values as follows. What’s the mean number of cars entering the freeway. Mean # Road Cars 1 800 2 1000 3 600 Mean of the Sum of Random Variables For any two random variables, X and Y, if = + then the expected value of T is = = + Ex: What is the standard deviation of the # of cars coming from each road onto the freeway. Road Mean # Cars St. Dev. 1 800 34.5 2 1000 42.8 3 600 19.3 Variance of the Sum of Random Variables For any two random variables, X and Y, if = + then the variance of T is x y x y 3x2 y Mean 20 24 st dev 5 3 x y x y 3x y Mean 20 24 st dev 5 3 x y x y 3x2 y Mean 20 24 st dev 5 3 x y x y 3x y Mean 20 24 st dev 5 3 Find: and 3x2 y 3x2 y x P(x) y P(y) 3 0.32 10 0.22 4 0.14 20 0.34 5 0.12 30 0.18 6 0.42 40 0.26 Homework Worksheet