: Let X denote the result of a random experiment with the following cumulative distribution function (cdf): 0, x <1.5 | 1/ 6 , 1.5<=x < 2 | 1/ 2, 2 <= x <5 | 1 ,x >= 5

Calculate ?(1 ? ≤ 6) and ?(2 ≤ ? < 4.5)

b. Find the probability mass function (pmf) of ?

d. If it is known that the result of the experiment is integer, what is the probability that the result is 2? e. If it is given that the result of this experiment is an integer and a fair coin is tossed the number of times that the die shows, find the probability of obtaining exactly one head.

Thank you

An important part of the customer service responsibilities of a cable company is the speed with which service troubles can be repaired. Historically, the data show that the likelihood is 0.75 that troubles in a residential service can be repaired on the same day. For the first five troubles reported on a given day:

a) What is the probability that all five will be repaired on the same day?

b) What is the probability that fewer than two troubles will be repaired on the same day?

c) What is the probability that at least three troubles will be repaired on the same day? d) Find the mean number of troubles repaired on the same day.

A survey shows that 30% of adults own stocks and 40% own mutual funds. A sample of 10 adults is chosen. Use binomial distribution to answer the following questions.

(a) (7 points) What is the probability that there will be less than 2 adults owning mutual funds?

            Answer: ___________________________

(b) (6 points) Determine the probability that there will be at least 2 adults owning stocks.

            Answer: __________________________________

(c) (6 points) Determine the variance of the number of adults not owning stocks.

            Answer: ________________________________

(d) (6 points) What is the probability of getting 5 or 6 adults owning mutual funds?

            Answer: ___________________________

The number of people arriving at an emergency room follows a Poisson distribution with a rate of 9 people per hour.

a) What is the probability that exactly 7 patients will arrive during the next hour?
b. What is the probability that at least 7 patients will arrive during the next hour?

c. How many people do you expect to arrive in the next two hours?

d. One in four patients who come to the emergency room in hospital.

Calculate the probability that during the next 2 hours exactly 20 people will arrive and less than 7 will be hospitalized

A local university reports that 3% of its students take their general education courses on a pass/fail basis. Assume that 50 students are registered for a general education course.

a. Define the random variable in words for this experiment.

b. What is the expected number of students who have registered on a pass/fail basis?

c. What is the probability that exactly 5 are registered on a pass/fail basis?

d. What is the probability that more than 3 are registered on a pass/fail basis?

e. What is the probability that less than 4 are registered on a pass/fail basis?

A University found that 27% of its graduates have taken an introductory statistics course. Assume that a group of 15 graduates have been selected.

1. Compute the probability that from this group, there are exactly 2 graduates that have taken an introductory statistics course.
2. Compute the probability that from this group, there are at most 3 graduates that have taken an introductory statistics course.
3. Compute the probability that from this group, there are at least 4 graduates that have taken an introductory statistics course.
4. Compute the expected number, the variance and the standard deviation of graduates that have taken an introductory statistics course.

Assume that when shopping for a Nintendo Switch, the probability of finding the Switch at any given store is .25. We’ll assume that this is modeled by a binomial distribution that uses the formula:

P(x successes in n trials) = nCx · p x · (1 − p) n−x Assuming that n = 9 people go shopping for the Switch,

(a) what is the probability that 3 people will find the switch?

(b) what is the probability that at least 2 people will find the Switch?

(c) what is the expected number (µ) of the 9 people that can find the Switch when going shopping?

A multiple-choice test has 6 questions. There are 4 choices for each question. A student who has not studied for the test decides to answer all questions randomly with a Probability of success = p =.25. Let X represent the number of correct answers out of six questions.

(i) Use binomial distribution to complete the table

(ii) Find the probability that the student will be successful in at least 4 questions.

(iii) Find the probability that the student will be successful in at most 3 problems

(iv) Compute the mean μ and the standard deviation σ for this problem.

A company that manufactures cans reports that the number of process interruptions due to mechanical problems in its assembly line is a Poisson random variable with an average rate 1.5 times per shift (8-hour shifts).
a) What is the probability that the line will have problems twice during the night shift?

b) What is the probability that the line can operate for 3 consecutive shifts without interruptions?

c) What is the probability that the line will operate without interruption on at least 3 of the 5 days in a randomly chosen week? (each day consists of 3 shifts)

A website manager has noticed that during the evening hours, about 3 people per minute check out from their shopping cart and make an online purchase. She believes that each purchase is independent of others and assumes that the distribution of number of purchases per minute on the website follows Poisson distribution.

1. What is the probability that during a 1-minute period, at least one purchase is made?

2. What is the probability that no one makes a purchase in next 2 minutes?

3. What is the probability that no more than 5 people will make a purchase in next three

   minutes?