STAT 400 Lecture AL1 Practice Problems 2 (Part 1) Spring 2015 Dalpiaz 1. Homer Simpson is going to Moe’s Bar for some Flaming Moe’s. Let X denote the number of Flaming Moe’s that Homer Simpson will drink. Suppose X has the following probability distribution: x f ( x ) 0 1 2 3 4 0.1 0.2 0.3 0.3 a) Find the probability f ( 4 ) = P(X = 4). b) Find the probability P(X ≥ 1). c) Find the probability P(X ≥ 1 | X < 3). d) Compute the expected value of X, E(X). e) Compute the standard deviation of X, SD(X). 1. (continued) Suppose each Flaming Moe costs $1.50, and there is a cover charge of $1.00 at the door. Let Y denote the amount of money Homer Simpson spends at the bar. Then Y = 1.50 ⋅ X + 1.00 . f) Find the probability that Homer would spend over $5.00. g) Find the expected amount of money that Homer Simpson would spend, E(Y). h) Find the standard deviation for the amount of money that Homer Simpson would spend, SD(Y).2. Each of three balls is randomly placed into one of three bowls. Let X denote the number of empty bowls. a) Find the probability distribution of X. “Hint”: f ( 0 ) = P ( X = 0 ) and f ( 2 ) = P ( X = 2 ) are easier to find. b) Find E ( X ). 3. Suppose we roll a pair of fair 6-sided dice. Let Y denote the difference between the outcomes on the two dice. Construct the probability distribution of Y and compute its expected value. 4. Honest Harry introduced a new “savings opportunity” at Honest Harry’s Used Car Dealership. A customer is offered the opportunity to roll four fair (balanced) 6-sided dice. If the customer rolls at least one “6”, Honest Harry takes an extra $1,000 off the price of the car. However, if the customer does not have a “6”, an extra $1,000 is added to the price of the car. What is the customer’s expected savings from accepting this “savings opportunity”? 5. Upon examination of the claims records of its policy holders over a period of three years, an insurance company obtained the following probability distribution of X = number of claims in three years: x f (x) 0 0.35 a) Calculate the average number of claims per policy holder, E( X ). b) Calculate the standard deviation of X, SD( X ). 1 0.30 2 0.20 3 0.10 4 0.056. 2.1-10 2.1-10 Suppose there are 3 defective items in a lot (collection) of 50 items. A sample of size 10 is taken at random and without replacement. Let X denote the number of defective items in the sample. Find the probability that the sample contains a) Exactly one defective item. b) At most one defective item. 7. Let the random variable X be the number of days that a certain patient needs to be in the hospital. Suppose X has the p.m.f. f ( x ) = 105x−, x = 1,2,3,4. a) Find E ( X ), the expected number of days the patient needs to be in the hospital. b) 2.2-3 2.2-5 If the patient is to receive $200 from the insurance company for each of the first two days in the hospital and $100 for each day after the first two days, what is the expected payment for the hospitalization? "Hint": If the patient spends three days in the hospital, the payment is $500. 8. 2.2-4 2.2-6 An insurance company sells an automobile policy with a deductible of one unit. Let X be the amount of the loss having pmf f ( x ) = ==.6,5,4,3,2,1,,0,9.0 xxcx where c is a constant. Determine c and the expected value of the amount the insurance company must pay.9. Mr. George Room proposed to Ms. Brittany Ride with a diamond ring valued at $8,000. Ms. B. Ride thinks the probability that it will be lost or stolen in a given year is .006. She can buy an Total insurance policy for one year for $50 that will completely reimburse the loss of the ring if it gets lost or stolen. She can also buy Regular insurance policy for one year for $30 that will reimburse 75% of the cost of the ring if it gets lost or stolen. a) Determine the payoff table. θ 1 ring is lost or stolen θ 2 ring is fine a 1 buy Total a 2 buy Regular a 3 do not buy insurance b) Compute the Expected Payoff for all three actions and determine the optimal action. 10. A credit card company examines its records to determine the length of time (in weeks) between the time a bill is sent and payment is received. The following table shows the results: Number of Weeks 1 2 3 4 5 Proportion 0.15 0.30 0.25 0.10 Let X denote the number of weeks between the time a bill is sent and payment is received. a) Find the missing proportion. b) Find P(X > 3). c) Find P(2 ≤ X ≤ 4). d) Find the expected (average) number of weeks between the time a bill is sent and payment is received, E(X). e) Find Var(X). f) Find SD(X).Answers: 1. Homer Simpson is going to Moe’s Bar for some Flaming Moe’s. Let X denote the number of Flaming Moe’s that Homer Simpson will drink. Suppose X has the following probability distribution: x f ( x ) x f ( x ) x 2 f ( x ) 0 1 2 3 4 0.1 0.2 0.3 0.3 0.1 0.0 0.2 0.6 0.9 0.4 0.0 0.2 1.2 2.7 1.6 1.0 2.1 5.7 a) Find the probability f ( 4 ) = P(X = 4). f ( 4 ) = 1 − [ 0.1 +0.2 + 0.3 + 0.3 ] = 0.10. b) Find the probability P(X ≥ 1). P(X ≥ 1) = 0.90. c) Find the probability P(X ≥ 1 | X < 3). P(X ≥ 1 | X < 3) = 60503)P(X3)X 1P(X..=<<∩≥ = 0.8333. d) Compute the expected value of X, E(X). E(X) = ∑⋅all x(x) x f = 2.1. e) Compute the standard deviation of X, SD(X). Var (X) = [ ]2all x2E(X)(x) x −⋅∑ f = 5.7 − (2.1)2 = 1.29. SD(X) = 29.1 = 1.1358.1. (continued) Suppose each Flaming Moe costs $1.50, and there is a cover charge of $1.00 at the door. Let Y denote the amount of money Homer Simpson spends at the bar. Then Y = 1.50 ⋅ X + 1.00 . f) Find the probability that Homer would spend over $5.00. x y f (x) = f (y) 0 1 2 3 4 $1.00 $2.50 $4.00 $5.50 $7.00 0.10 0.20 0.30 0.30 0.10 1.00 P(Y > $5.00) = P(X ≥ 3) = 0.40. g) Find the expected amount of money that Homer Simpson would spend, E(Y). µ Y = E(Y) = 1.50 ⋅ E(X) + 1.00 = $4.15. (On average, Homer drinks 2.1 Flaming Moe’s per visit, his expected payment for the drinks is $3.15. His expected total payment is $4.15 since he has to pay $1.00 …
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