The manager of the commercial mortgage department of a large bank has collected data during the past two years concerning the number of commercial mortgages approved per week. The results from these two years ​(104 weeks) are shown to the right.

a. Compute the expected number of mortgages approved per week.

b. Compute the standard deviation.

c. What is the probability that there will be more than one commercial mortgage approved in a given​ week?

Number_Approved   Frequency
0   12
1   25
2   32
3   18
4   9
5   5
6   2
7   1

The expected number of mortgages approved per week is

​(Round to three decimal places as​ needed.)

b. The standard deviation is

​(Round to three decimal places as​ needed.)

c. The probability that there will be more than one commercial mortgage approved in a given week is

​(Round to three decimal places as​ needed.)

1. An IKEA “Tarva” bed frame is assembled with screws and Allen wrenches. The screws and wrenches for the Tarva kits are grabbed at random from large bins at the factory by two different people who never interact. Based on several years of data, it is known that 95% of Tarvas come with the proper size Allen wrenches, and 85% of them come with the correct number of screws. Hints for the two problems below: It may help to write out the list of all possible outcomes of this random process. Also, remember that the probabilities of outcomes add, and that independent probabilities multiply.

   1. (a) The bed frame can only be assembled if it contains the proper size Allen wrench and the correct number of screws. What is the probability that your bed frame can be assembled?

   2. (b) What is the probability that you have either the proper size wrench or the correct number of screws, but not both?

I want question 8 answered question 7 is posted because data from that question is required to answer 8

7. The following is the joint probability distribution of number of car crashes (C) and car make (M). C = 0 C = 1 C = 2 C = 3 C = 4 TOYOTA (M = 0) 0.35 0.065 0.05 0.025 0.01 OTHER (M = 1) 0.45 0.035 0.01 0.005 0.00 A. Report the marginal probability distribution for C B. What is the average number of car crash? C. What is the variance of the number of crashes? D. Calculate σCM and ρCM.

8. Suppose car manufacturers are penalized (P) on the basis of the following formula P = 60,000 + 6C – 2M Using your answers for Question 7, calculate the following A. The average penalty (P) B. The variance of penalty (P)

An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X = the number of points earned on the first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table.

(a) what is the probability that a randomly selected student scores 5 on both parts?
(b) what is the probability that a randomly selected student scores at least 5 on part 1 and no points on part 2?

(c) find the marginal PMFs of X and Y (find PMF of X and PMF of Y).

(d) If the score recorded in the grade book is the toal number of points earned in two parts, what is the expected recorded score E(X+Y)?

2) A sample of customers for Montana Gas and Electric resulted in the following distribution of monthly charges.

a) What is the probability that a customer has monthly electric cost of $150 or more?

b) What is the probability of a customer having a monthly electric cost of less than $150?

In a football or soccer game, you have 22 players, from both teams, in the field. What is the probability of having any two players with the same birthday? (just assume 365 days a year and don’t have to do the exact calendar month and day, use the day number from 1 to 365)

Find the closed form mathematical solution by probability theory. Show your derivation/proof.

Pete is doing a science-experiment and have decided to work on it until it succeeds. The chance of success on any given day is 0.001. Let X be the number of days until he succeeds. Which probability distribution does X have? What is E(X)? What is the probabilty of Pete succeeding in his first year? If he doesn't succeed the first year, what's the probability of success in the second year?

7. (9 pts.) On weekend nights, a large urban hospital has an average of 4.8 emergency arrivals per hour. Let X be the number of arrivals per hour on a weekend night at this hospital. Assume that successive arrivals are random and independent.

(a ) What is the distribution of X ?

(b) What is the probability of observing exactly two emergency arrivals?

(c) What is the probability of observing more than two emergency arrivals?

12. What is the probablity of obtaining a number higher than 4 when rolling a dice?

13.

a. What is the probability of obtaining a total of 10 or 12 when launching 2 dice at the same time (adding total of both dices)?

b. ¿Does it change if you add a 3rd dice? How?

14) What would be the probability of obtaining a total of 2 if you roll 3 dice at the same time?