1. Consider a single server queueing system. Arrivals take place every four minutes (first arrival takes place at t=0). Service times are random and take the value of 1 with probability 1/2 and 5 with probability ½. Assume that the following service time sequence is generated: S1=5, S2=5, S3=5, S4=1, S5=5. Simulate for T=20 minutes and estimate the average number of customers in the system and the average waiting time of a customer.

In many countries around the world, couples look to a son to take care of them in their old age. They are inclined to keep having children until they have a son. For this problem, imagine that a government considers permitting a couple to continue having children until they have a son. Assume that the birth of a male is as likely as the birth of a female.
(a) What’s the probability that a couple will have only one child?
(b) What’s the probability that a family will have two children?
(c) What would be the average number of children per family?

Find the indicated probabilities using the geometric​ distribution, the Poisson​ distribution, or the binomial distribution. Then determine if the events are unusual. If​ convenient, use the appropriate probability table or technology to find the probabilities.

Fifty-nine percent of adults say that they have cheated on a test or exam before. You randomly select six adults. Find the probability that the number of adults who say that they have cheated on a test or exam before is​ (a) exactly four​,​(b) more than two​, and​ (c) at most five.

The lifetime of a certain battery is normally distributed with a mean value of 20 hours and a standard deviation of 2.5 hours.

a. What are the distribution parameters (μ and σ) of the sample mean if you sample a four pack of batteries from this population?

b. If there are four batteries in a pack, what is the probability that the average lifetime of these four batteries lies between 18 and 20?

c. What happens to the probability in part 1.b. if the number of batteries in the sample goes up?

Show all work please

A Geiger counter counts the number of alpha particles from radioactive material. Over a long period of time, an average of 25 particles per minute occurs. Assume the arrival of particles at the counter follows a Poisson distribution.

Find the probability that at least one particle arrives in a particular one second period. Round your answer to four decimals.
   
Find the probability that at least two particles arrive in a particular 2 second period. Round your answer to four decimals.
   

Consider two assets, A and B, in a competitive market populated by a large number of risk-averse
agents.
ForassetA: pricepA =10attime0;andpaysXA =12attime1.
For asset B: price pB = 10 at time 0; and pays, at time 1, XB = 20 with probability 0.5, and XB = 4 with probability 0.5.
a. Please calculate the expected return and the standard deviation of return for asset A. b. Please calculate the expected return and the standard deviation of return for asset B. c. Is pA = pB reasonable? Please explain.

A study conducted by the MLC on campus found the number of hours spent at the MLC during a week. The distribution of times spent at the MLC is normally distributed with an average of 75 minutes and a standard deviation of 18 minutes.

(a) If a single student that uses the MLC is selected at random, what is the probability that this student spent between 80 and 100 minutes at the MLC that week?

(b) Now if a sample of 20 students is selected, what is the probability that the sample will have a mean time spent at the MLC that is between 80 and 100 minutes?

Each day John performs the following experiment. He flips a fair coin repeatedly until he sees a T and counts the number of coin flips needed.

(a) Approximate the probability that in a year there are at least 3 days when he needed more than 10 coin flips. Argue why this approximation is appropriate.

(b) Approximate the probability that in a year there are more than 50 days when he needed exactly 3 coin flips. Argue why this approximation is appropriate.

An Olympic archer is able to hit a bulls-eye 80% of the time. Assume each shot is independent of the others. If she shoots 7 arrows, what is the probability of each result described below? a. What is the expected number of bulls-eyes for the 7 attempts? b. What is the standard deviation? c. She gets at least 4 bulls-eyes? d. Make a histogram showing the probability that r = 0, 1, 2, 3, 4, 5, 6, 7 bulls-eyes?

A) A certain area is hit by 6 Heat Waves a year on average. Find the probability that in a given year that area will be hit by fewer than 5 Heat Waves (5 is not included).

a) 0.2851

b) 0.4457

c) 0.1339

d) 0.1606

B) The probability of a successful rocket launch is 0.9. Test launches are conducted until three successful launches are achieved. Let X be the total number of launches needed. What kind of distribution does X follow?

C) What are the properties of a Bernoulli process?