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1 . At a certain gas station, 4% of the customers use regular unleaded gas ( A ), % use extra unleaded gas ( A ), and % use premium unleaded gas ( A ). Of those customers using regular gas, onl % fill their tanks (event B). Of those customers using extra gas, 6% fill their tanks, whereas of those using premium, % fill their tanks. a. What is the probabilit that the next customer will request extra unleaded gas and fill the tank? b. What is the probabilit that the next customer fills the tank? c. If the next customer fills the tank, what is the probabilit that regular gas is requested? Extra gas? Premium gas? ANSWER: P( A ) =.4, P( A ) =., ( ) P(B A ) =. P(B A ) =.6 P(B A ) =. Therefore, P A =. ( ) ( ) ( ) ( )( ) P A B = P A P B A =.4. =. ( ) ( ) ( ) ( )( ) P A B = P A P B A =..6 =. ( ) ( ) ( ) ( )( ) P A B = P A P B A =.. =. a. ( A B) =. b. P(B) = PA ( B) + PA ( B) + PA ( B) =.+.+.=.4 PA ( B). c. P( PAB ( ) = = =.64 PB ( ).4 PA ( B). PAB ( ) = = =.46 PB ( ).4 PA ( B). = == PB ( ).4 ( ) =.7 PAB

2 . The number of tickets issued b a meter reader for parking meter violations can be modeled b a Poisson process with a rate parameter of five per hour. a. What is the probabilit that exactl three tickets are given out during a particular hour? b. What is the probabilit that at least three tickets are given out during a particular hour? c. How man tickets do ou expect to be given during a 4 min period? ANSWER: a. PX ( = ) = F(;) F(;) =.6. =.4 b. PX ( ) = PX ( ) = F(;) =. =.87 c. Tickets are given at the rate of per hour, so for a 4 minute period the rate is λ = ()(.7) =.7, which is also the expected number of tickets in a 4 minute period.. An aircraft can seat passengers, and each of the passengers booked on the flight has a probabilit of.9 of actuall arriving at the gate to board the plane, independent of the other passengers. a. Suppose the airline books passengers on the flight. What is the probabilit that there will be insufficient seats to accommodate all of the passengers who wish to board the plane? b. If the airline wants to be 7% confident that there will be no more than passengers who wish to board the plane, how man passengers can be booked on the flight? 4. In commuting to school, I must first get on a bus near m house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with A = and B =, then it can be shown that m total waiting time Y has the pdf f ( ) = < < or > a) Sketch a graph of the pdf of Y

3 b) Verif that f ( ) d = c) What is the probabilit that the total waiting time is at most minutes? d) What is the probabilit that the total waiting time is between and 8 minutes? e) What is the probabilit that the total waiting time is either less than minutes or more than 6 minutes? a...4. f(x)... x b. f ( ) d = d + ( ) d = + (4 ) ( ) = + = = + 9 c. P(Y ) = d =. 8 = d. P( Y 8) = P(Y 8) P(Y < ) = = =. 74 e. P(Y < or Y > 6) = d + = =. 4 d 6. Consider the pdf for total waiting time Y for two buses f ( ) = < < or > a) Compute and sketch the cdf of Y b) Compute E(Y) and Var(Y)

4 a. For, F() = udu = For, F() = f ( u) du = f ( u) du + f ( u) du = + u du =. F(x).. x b. E(Y) = b straightforward integration (or b smmetr of f()), and V(Y)= = Let X denote the number of Son digital cameras sold during a particular week b a certain store. The pmf of X is x 4 P(X = x)..... Sixt percent of all customers who purchase these cameras also bu an extended warrant. Let Y denote the number of purchasers during this week who bu an extended warrant. a) What is the value of P(X = 4, Y = )? b) Calculate P(X = Y) c) Determine the joint pmf of X and Y d) Determine the marginal pmf of Y 4 a. p(4,) = P( Y = X = 4) P(X = 4) = (.6) (.4) (.) =. 8 b. P(X = Y) = p(,) + p(,) + p(,) + p(,) + p(4,4) =.+(.)(.6) + (.)(.6) + (.)(.6) + (.)(.6) 4 =.44

5 c. p(x,) = unless =,,, x; x =,,,, 4. For an such pair, x x p(x,) = P(Y = X = x) P(X = x) = (. 6) (.4) px ( x) d. p (4) = p( = 4) = p(x = 4, = 4) = p(4,4) = (.6) 4 (.) =.94 4 p () = p(,) + p(4,) = (.6) (.) + (.6) (.4)(.) =. 8 p () = p(,) + p(,) + p(4,) = (.6) (.) + (.6) (.4)(.) 4 + (.6) (.4) (.) =.678 p () = p(,) + p(,) + p(,) + p(4,) = (. 6)(.) + (.6)(.4)(.) 4 (.6)(.4) (.) + (.6)(.4) (.) =.9 p () = [ ] = Somchai and Soming have agreed to meet between : PM and 6: PM for dinner at a local health food restaurant. Let X be Somchai s arrival time and Y be Soming s arrival time. Suppose X and Y are independent with each uniforml distributed on the interval [, 6]. a) What is the joint pdf of X and Y? b) What is the probabilit that the both arrive between : and :4 PM? c) If the first one to arrive will wait onl minutes before leaving to eat elsewhere, what is the probabilit that the have dinner at the health food restaurant? x 6, 6 a. f(x,) = otherwise since f x (x) =, f () = for x 6, 6 b. P(. X.7,. Y.7) = P(. X.7) P(. Y.7) = (b independence) (.)(.) =. c. 6 I = x+/6 = x / 6 II 6

6 P((X,Y) A) = A dxd = area of A = (area of I + area of II ) = = = A nut compan markets cans of deluxe mixed nuts containing almonds, cashews, and peanuts. Suppose the net weight of each can is exactl lb., but the weight contribution of each tpe of nut is random. Because the three weighs sum to, a joint probabilit model for an two gives all necessar information about the weight of the third tpe. Let the joint pdf for (X, Y) be f X, Y 4x ( x, ) = x, otherwise, x + Are X and Y correlated? If not, what s the correlation coefficient of X and Y? Cov(X,Y) = and μ x = μ =. 7 E(X ) = x f x ( x) dx = x ( x dx) = =, 6 4 so Var (X) = = Similarl, Var(Y) =, 7 so ρ X, Y = = =

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