Customers arrive in a certain shop according to an approximate Poissonprocess on the average of two every 6 minutes.

(a) Using the Poisson distribution calculate the probability of two or more customersarrive in a 2-minute period.

(b) Consider X denote number of customers and X follows binomial distribution withparametersn= 100. Using the binomial distribution calculate the probability oftwo or more customers arrive in a 2-minute period.

(c) Let Y denote the waiting time in minutes until the first customer arrives. (i) Whatis the pdf ofY? (ii) Findq1=π0.75

(d) Let Y denote the waiting time in minutes until the first customer arrives. What isthe probability that the shopkeeper will have to wait more than 3 minutes for thearrival of the first customer ?

(e) What is the probability that shopkeeper will wait more than 3 minutes before bothof the first two customers arrive?

Suppose an audit process is taking place at a small airport, and the head
auditor arrives randomly throughout the day and assesses the operation of the airport.
She shows up in one-hr intervals at a time. The time between aircraft arrivals at this
small airport is exponentially distributed with a mean of 75 minutes. You may assume
the operation of the small airport is independent of the time interval under consideration.
a) What is the probability that at least four aircrafts arrive within an hour?
b)What is the probability that more than three and less than six aircrafts arrive within an
hour?
C)What is the amount of time (in hours) such that the probability of aircraft arrivals in the
interval is 75%?
D) What is the probability that the third interval the auditor shows up for is the second
interval with at least four aircraft arrivals?
e) If the auditor decides to show up four different times in a day, what is the mean of the
number of intervals she will experience with less than four aircraft arrivals?

5.         Wellington Fabrics of New Zealand produces bolts of woolen cloth for export. Each bolt contains 30 yards of fabric. Industry standards call for the average number of defects per fabric bolt to not exceed five. An inspector randomly selected a bolt of cloth, examined the first 3 yards and found 3 defects therein. The company assumes that the defect rate follows the Poisson distribution.

a.          Given the above information, if the company is meeting the industry standards, what is the average number of defects expected in a 3-yard segment of a cloth bolt?

b.         Calculate the probability of finding 3 or more defects in a 3-yard segment of cloth given the above information.

c.          Given your answer to part b., does it appear that the company is meeting the industry standard for quality? Explain briefly.

d.         To verify the findings the inspector examines 15 yards of another bolt of cloth and discovers five defects.

i.          Calculate the average number of defects expected in a 15-yard segment of cloth;

ii.         Calculate the probability of finding five or more defects in the 15-yard cloth segment.

e.          The normal distribution can be used to approximate the Poisson distribution. To calculate a z-score you need the mean and standard deviation. Remember that the mean and variance of a Poisson random variable are equal.

i.          Calculate the mean and standard deviation for the number of defects in a 15-yard segment of cloth.

ii.         Use this mean and standard deviation to calculate the probability of finding five or more defects in a 15-yard segment of cloth using the normal approximation. [Remember to use the continuity correction!]

iii.        How accurate is this approximation?

(a) Suppose a dealer draws one card from a standard, properly shuffled, 52-card deck of cards (that is, all Jokers have been removed). (i) Describe the sample space - i.e. all of the possible outcomes (no need to write them all out, just describe them in words). (1 point) (ii) What is the probability that the card is a Heart? (1 point) (iii) What is the probability that the card is a 6? (1 point) (iv) What is the probability that the card has a number on the face (i.e. not Jack, Queen, King, or Ace)? (1 point)

(b) Suppose I flip a fair coin twice in a row. (i) Write out the set of possible outcomes (the sample space). (1 point) (ii) What is the probability that at least one flip lands with Heads facing up? (1 point) (iii) What is the probability that both flips land with Heads facing up? (1 point)

(c) Consider a business owner who can make one of three decisions about a new product. Each decision generates a lottery over different possible revenue outcomes. If he makes Decision A, then the resulting lottery, p A, generates $20, 000 in revenue with probability 0.3, $12, 000 in revenue with probability 0.15, $5, 000 in revenue with probability 0.35, and $2, 000 with probability 0.2. If he makes Decision B, then the resulting lottery, p B, generates $30, 000 in revenue with probability 0.1, $10, 000 in revenue with probability 0.6, and $1, 000 with probability 0.3. If he makes Decision C, then the resulting lottery, p C , generates $20, 000 in revenue with probability 0.2, and $8, 000 with probability 0.8. (i) Write each lottery in the form (p1, x1; p2, x2; ...; pn, xn). (2 points) (ii) Calculate the expected value (of revenue) from each decision. (3 points)

Refer to the given consumption schedules. DI signifies disposable income and C represents consumption expenditures. All figures are in billions of dollars. At an income level of $40 billion, the average propensity to consume

On level ground a shell is fired with an initial velocity of 36.0 m/s at 58.0 ∘ above the horizontal and feels no appreciable air resistance. ((INCLUDE ALL ANSWER WITH THE UNITS LIKE m/s OR WHATEVER)) THANK YOU

A) Find the horizontal and vertical components of the shell's initial velocity.

B) How long does it take the shell to reach its highest point?

C) Find its maximum height above the ground.

D) How far from its firing point does the shell land?

E) At its highest point, find the horizontal and vertical components of its acceleration.

F) At its highest point, find the horizontal and vertical components of its velocity.

Argon gas is initially at a pressure of 10 bar and temperature of 45oC (state 1), while occupying a volume of 0.8 m3 in a frictionless piston-cylinder arrangement. The gas then undergoes a reversible constant volume process to a pressure of 6 bar (state 2), followed by a constant pressure process in which the temperature is restored to 45oC (state 3). a) Sketch the processes on T-s and p-v diagrams b) Assuming perfect gas behaviour determine i) The work done by each process. ii) The heat transfer during each process. iii) The change in entropy in process state 3 to 1. (For Argon take Cp = 519.6 J.kg^-1K^-1 and relative molar mass M=40

Prob. 4-68 A piston–cylinder device contains 2.2 kg of nitrogen initially at 100 kPa and 25C. The nitrogen is now compressed slowly in a polytropic process during which PV^1.3 = constant until the volume is reduced by one-half. Determine the work done and the heat transfer for this process.

4–69    Reconsider Prob. 4–68. Using EES (or other) software,plot the process described in the problem on a P-V diagram, and investigate the effect of the poly- tropic exponent n on the boundary work and heat transfer. Let the polytropic exponent vary from 1.0 to 1.4. Plot the boundary work and the heat transfer versus the polytropic exponent, and discuss the results.

Consider an airplane patterned after the Beechcraft Bonanza V-tailed, single-engine light private airplane. The characteristics of the airplane are as follows: aspect ratio = 6.2, wing area = 181 ft², Oswald efficiency factor = 0.91, weight = 3500 lb, and zero-lift drag coefficient = 0.027. The airplane is powered by a single piston engine of 345 hp maximum at sea level. Assume the power of the engine is proportional to free-stream density. The two-blade propeller has an efficiency of 0.83. Estimate the sea-level liftoff distance for the airplane. Assume a paved runway and C_L,max = 1.1 during the ground roll. When the airplane is on the ground, the wings are 4 ft above the ground.

Consider a log splitter with a load capacity of 15 ton which is run by a 10 GPM hydraulic pump. The bore and rod diameter of the log splitter cylinder are 4 in. and 2 in., respectively, whereas the stroke length is 24 in. Determine the following:

a) Maximum pressure generated in the extension stroke

b)Extension velocity of the cylinder

c)Power (HP) generated in the extension stroke when pushing the maximum load

d) Retraction velocity of the cylinder

e) Power (HP) generated in retraction stroke when pulling the maximum load

f) % increase in velocity of the piston during retraction compared to extension

g) Cycle time of the cylinder (Hint: Cycle time = Extension time + Retraction time)