The number of customers that visit a 24-hour local supermarket during the off-peak hours between 8am - 11am is known to be distributed as Poisson with mean 2 per hour. Determine the probability that

i. no customers will visit the supermarket between 8am - 9am.

ii. exactly one customer will visit the supermarket between 9am - 11am.

iii. exactly one customer will visit the supermarket for each of the time intervals 8am - 9am and 9am-11am.

A critical component on a submarine has an operating lifetime that is exponentially distributed with mean 0.50 years. As soon as a component fails, it is replaced by a new one having statistically identical properties. What is the smallest number of spare components that the submarine should stock if it is leaving for a one-year tour and wishes the probability of having an inoperable unit caused by failures exceeding the spare inventory to be less than 0.02?

Please show that step by step and follow the comment

A manufacturer of nickel-hydrogen batteries randomly selects 100 nickel plates for test cells, cycles them a specified number of times, and determines that 14 of the plates have blistered. If it is really the case that 15% of all plates blister under these circumstances and a sample size of 100 is used, how likely is it that the null hypothesis of part (a) will not be rejected (to 4 decimal places). (The probability of making a type 2 error when the true value of p = 0.15)

A report at LinkedIn website reports that on average people will have 15 jobs over their lifetime. Suppose the standard deviation for how many jobs people have is 3.6 jobs. Assuming these figures are correct, Consider a representative sample of 80 retired individuals.

a. Give the properties of the sampling distribution for the sample mean number of jobs over their lifetime. (8)

b. What is the probability the sample mean will be more than 14 jobs per person (10)

Suppose that the time between two absences of a supermarket employee in a company follows an exponential distribution. Based on the past data on number of supermarket employees absent in a particular month, it has been observed that on an average 5 supermarket employees tend to be absent in any given month. What is the probability that the time between two absences for a particular supermarket employee is:

1. Less than 2 weeks

2. Between 2 weeks and 4 weeks

3. More than a month

Paul is the pharmacist at RiteAid pharmacy where customers arrive on average every 10 minutes (Exponential distribution). Paul can serve on average 8 customers per hour (Poisson distribution).

Using Queuing theory, calculate

a. the average number of customers waiting in line

b. the average waiting time in line

c. the average waiting time in the system (line + order filling)

d. the system utilization

e. the probability that no customers are in the pharmacy

An e-retailing company assigns an ID to online orders received. Order ID starts with 3-digit number, followed by four letters of alphabet.

a) How many unique order IDs could be generated?

b) What is the probability that first order is assigned an ID where sum of 3 digits equals 4? For instance, consider order ID “123 ABCD” with sum of three digits equals 1+2+3 = 6.

For a particular intersection, it has been bserved that the rate of people going thorugh the intersection while talking on their cell phone follows a Poisson process with an average of 6 people talking on their cell ohone per minute. Let X = number of people talking on their cell phone while driving through this intersection every 5 minutes. What is the probability that the 15th person talking on their cell phone will be observed going through this intersection in at most 3 minutes?

There are two urns that, between them, contain five balls. At each time step, one of the five balls is moved to the other urn. Let the state variable be the number of balls in Urn 1. Find the fixed vector.

a) Draw a state transition diagram and find the transition matrix.

b) Is this a regular chain? Is this an ergodic chain?

c) Find the fixed vector. What is the probability that in the long run Urn 1 has at most one ball?

There are 10 democrats and 8 Republicans on a senate committee. From this group a 5- member subcommittee is to be formed. Find the number of 5-person subcommittee that consists of:

a)Any members of the senate committee.

b)Democrats only.

c)3 Republicans and 2 Democrats.

d)At least 4 democrats.

e)John Edwards,who is a Democrat, and any 4 Republicans

f)What is the probability that the 5-members subcommittee will include 3 Republicans and 2 Democrats?