Repair calls are handled by one repairman at a photocopy shop. Repair time, including travel time, is exponentially distributed, with a mean of 2.7 hours per call. Requests for copier repairs come in at a mean rate of 1.8 per eight-hour day (assume Poisson)

a. Determine the average number of customers awaiting repairs. (Round your answer to 2 decimal places.) Number of customers

b. Determine system utilization. (Round your answer to 2 decimal places. Omit the "%" sign in your response.) System utilization %

c. Determine the amount of time during an eight-hour day that the repairman is not out on a call. (Round your answer to 2 decimal places.) Amount of time hours

d. Determine the probability of two or more customers in the system. (Do not round intermediate calculations. Round your answer to 4 decimal places.) Probability

The first significant digit in any number must be​ 1, 2,​ 3, 4,​ 5, 6,​ 7, 8, or 9. It was discovered that first digits do not occur with equal frequency. Probabilities of occurrence to the first digit in a number are shown in the accompanying table. The probability distribution is now known as​ Benford's Law. For​ example, the following distribution represents the first digits in 226 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.

Using the table below and a significance level of a=0.01, complete part​ (a) below.

(a) What is the test​ statistic? (round to three decimal places as needed)

A small town with one hospital has 4 ambulances to supply ambulance service. Requests for ambulances during nonholiday weekends average 0.68 per hour and tend to be Poisson-distributed. Travel and assistance time averages 2.40 hours per call and follows an exponential distribution Use Table 1.

a. Find system utilization. (Round your answer to the nearest whole percent. Omit the "%" sign in your response.)

System utilization             %

b. Find the average number of customers waiting. (Round your answer to 3 decimal places.)

Average number of customers            

c. Find the average time customers wait for an ambulance. (Round your answer to 3 decimal places.)

Average time             hour

d. Find the probability that all ambulances will be busy when a call comes in. (Round intermediate calculations to 3 decimal places and final answer to 2 decimal places.)

Probability  

The problem of matching aircraft to passenger demand on each flight leg is called the flight assignment problem in the airline industry. Suppose the demand for the 6 p.m. flight from Toledo Express Airport to Chicago's O'Hare Airport on Cheapfare Airlines is normally distributed with a mean of 144 passengers and a standard deviation of 42. Round probabilities in parts (a) through (c) to four decimal places. a) Suppose a Boeing 757 with a capacity of 186 passengers is assigned to this flight. What is the probability that the demand will exceed the capacity of this airplane? b) What is the probability that the demand for this flight will be at least 94 passengers but no more than 200 passengers? c) What is the probability that the demand for this flight will be less than 100 passengers? Round answers in parts (d) and (e) to the nearest whole number. d) If Cheapfare Airlines wants to limit the probability that this flight is overbooked to 1%, how much capacity should the airplane that is used for this flight have? passengers e) What is the 65th percentile of this distribution?

The distribution function W(x) for a one dimensional random walk gives the probability that an object moving randomly in the +x and -x directions is displaced by a distance x after N jumps, each jump of length l. The displacements are normally distributed which means the distribution function is W(x)=(1/sqrt2πσ^2)e^(−(x−<x>)^2)/2σ^2). In the displacement function, <x> is the average displacement <x>=Nl(p−q) and the variance is σ^2=<x^2>−<x>^2=4pqNl2σ^2. The parameter p is the probability of a displacement in the +x direction and q is the probability of a displacement in the -x direction. The variance σ^2 is a measure of the dispersion of the displacements.

The square root of the variance is the standard deviation σ. What is the probability that after N=20 jumps the organism will be found between x=0.00mm and x=0.224mm? Give your answer as a number between 0 and 1?

Suppose 4 Bernoulli trials, each with success probability p, are conducted such that the outcomes of the 4 experiments are mutually independent. Let the random variable X be the total number of successes over the 4 Bernoulli trials.

(a) Write down the sample space for the experiment consisting of 4 Bernoulli trials (the sample space is all possible sequences of length 4 of successes and failures you may use the symbols S and F).

(b) Give the support (range) X of X.

(c) Tabulate for X the probability distribution P_x(X = x) for x ∈ X in terms of the success probability p.

(d) Give the cdf F_x of X (Remember that F_x(x) must be defined for all x ∈ R) in terms of the success probability p.

(e) Make an accurate drawing of the cdf F_x of X in the case p = 1/2.

(f) State whether X is a discrete or a continuous rv and why.

The average number of miles driven on a full tank of gas in a certain model car before its​ low-fuel light comes on is 341. Assume this mileage follows the normal distribution with a standard deviation of 39 miles. Must show ALL calculations by use of formulas ONLY, no Excel use.

Complete parts a through d below.

a. What is the probability​ that, before the​ low-fuel light comes​ on, the car will travel less than 369 miles on the next tank of​ gas?

b. What is the probability​ that, before the​ low-fuel light comes​ on, the car will travel more than 248 miles on the next tank of​ gas?

c. What is the probability​ that, before the​ low-fuel light comes​ on, the car will travel between 259 and 279 miles on the next tank of​ gas?

d. What is the probability​ that, before the​ low-fuel light comes​ on, the car will travel exactly 279 miles on the next tank of​ gas?

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

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

(b) Consider X denote number of customers and X follows binomial distribution with parameters n= 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) What is the pdf ofY? (ii) Find q1=π0.75

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

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

A chain of motels had adopted a policy of giving a 3% discount to customers who pay in cash rather than by credit cards. Its experience is that 30% of all customers take the discount. Let Y = the number of discount takers among the next 20 customers.

7) What kind of probability distribution is appropriate in this case?
a) Normal Distribution b) Binomial Distribution c) Poisson Distribution d) Unknown Discrete Probability Distribution

8) Find the probability that exactly 5 of the next 20 customers will take the discount.

a) 0.17 b) 0.20 c) 0.30 d) 0.41

9) Find the probability that 5 or more customers will take the discount.

a) 0.17 b) 0.58 c) 0.76 d) 0.50

10) Find the expected value of Y.

a) 0.6 b) 5 c) 4 d) 6

11) Find the standard deviation of Y.

a) 4.2 b) 6 c) 3 d) 2.04