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.

Q5.In a waiting line situation, arrivals occur at a rate of 2 per minute, and the service times average 18 seconds. Assume the Poisson and exponential distributions [Hint: convert service time 18 seconds to service rate].JosiahA

a.   What is l?

b.   What is µ?

c.   Find probability of no units in the system.

d.   Find average number of units in the system.

e.   Find average time in the waiting line.

f.    Find average time in the system.

g.   Find probability that there is one person waiting.

h.   Find probability an arrival will have to wait.

IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose one individual is randomly chosen. Let X = IQ of an individual.

Part B

Find the probability that the person has an IQ greater than 110.

What is the probability? (Round your answer to four decimal places.)

Part C

The middle 60% of IQs fall between what two values?

State the two values. (Round your answers to the nearest whole number.)

What is the probability? (Round your answer to four decimal places.

F6: In a waiting line situation, arrivals occur around the clock at a rate of six per day, and the service occurs at one every three hours. Assume the Poisson and exponential distributions.

Please show your work

The distribution of the number of people in line at a grocery store has a mean of 3 and a variance of 9. A sample of the numbers of people in line in 50 stores is taken.

(a) Calculate the probability that the sample mean is more than 4? Round values to four decimal places.

(b) Calculate the probability the sample mean is less than 2.5. Round answers to four decimal places.

(c) Calculate the probability that the the sample mean differs from the population mean by less than 0.5. Round answers to four decimal places.

Please help, and show step by step. Thank you.

A sushi place in the food court is trying to determine how many bento boxes to make everyday. The price of a box is $12. The cost of making one box is $8. In the end of the day, any unsold boxes will be thrown away with no salvage value. Customer demand during the day is random and follows a certain probability distribution. What is the probability of stockout if the optimal order quantity is used?

Note 1: Keep 2 decimal places for your answer.

Note 2: Provide your answer as a number between 0 and 1. For example, if the probability is 0.1253, enter 0.13.

2. Show that the first derivative of the the moment generating function of the geometric evaluated at 0 gives you the mean.

3. Let X be distributed as a geometric with a probability of success of 0.10.

a. Give a truncated histogram (obviously you cannot put the whole sample space on the x-axis of the histogram) of this random variable.

b. Give F(x)

c. Find the probability it takes 10 or more trials to get the first success.

d. Here is a challenge. What is the probability that it takes an even number of trials to get the first success, i.e., P(X=2,4,6,8,...)

A and B toss a pair of coins in turn, with A tossing first. A's objective is to obtain TT and B's is to obtain TH (or HT). The game ends when either player reaches his or her objective, and that player is declared the winner. Assuming that H appears with probability p=1/3 in each coin.

1) Find the probability that A is the winner

2) Find the expected number of tosses of the coins

1. For each of the phenomena described below, propose a probability distribution for the numerical variable(s) involved and give the corresponding formula.
e) Compute the probability of observing exactly k alternations of colours in a drawing of 6 balls, with replacement, from an urn containing an equal number of red and blue balls. (Examples: RBBBBB is one alternation, whereas RBBRBB and RRBRBB each have three alternations.)

A lot of 100 washers contains 5 in which the variability in thickness around the circumference of the washer is unacceptable. A sample of 10 washers is selected at random, without replacement. (Round your answer to four decimal places.)

(a) What is the probability that none of the unacceptable washers is in the sample?

(b) What is the probability that at least one unacceptable washer is in the sample?

(c) What is the mean number of unacceptable washers in the sample?