Let A1, A2 and A3 be events except with respective probabilities 1/6 , 1/5, and 1/4.Let N be the number of these events that occur.

a) Write down a formula for N in terms of indicators.

b) Find E(N).



In each of the following cases, calculate Var(N):

c) A1, A2, A3 are disjoint;

d) they are independent;

e) A1 is in A2 is in A3.

Studd Enterprises sells big-screen televisions. A concern of management is the number of televisions sold each day. A recent study revealed the number of days that a given number of televisions were sold.

                        # of TV units sold      # of days

                                       0                             2

1.                         4

2.                       18

3.                       12

4.                       10

5.                         4

Answer the questions below. For each part, show your calculations and/or explain briefly how you arrived at your answer, as appropriate or needed.

Required:

1. Convert the frequency distribution above into a probability distribution (or relative frequency distribution) showing the proportion of days (rather than the number of days) that the number of televisions sold was 0, 1, 2, 3, 4, and 5 respectively. ( 3 marks)

2. Compute the mean of this general discrete probability distribution. ( 3 marks)

3. Compute the standard deviation of this general discrete probability distribution. ( 5 marks)

4. What is the probability that exactly 4 televisions will be sold on any given day? ( 1 mark)

5. What is the probability that 2 or more televisions will be sold on any given day? ( 1 mark)

6. What is the probability that less than 2 televisions will be sold on any given day? (1 mark)

7. What is the probability that between 1 and 4 televisions inclusive will be sold on any given day? (1 mark)

14. A coin is flipped 100 times, and 59 heads are observed. Find a 80% confidence interval of π (the true population proportion of getting heads) and draw a conclusion based on the collected data.

Find the P-Value of the test. Ha: π =1/2. Vs. Ha: π ≠1/2.

A) Less than 1%.

B) Between 1% and 2%

C) Between 2% and 3%

D) Between 3% and 4%

E) Between 4% and 5%

F) Between 5% and 7%

G) Between 7% and 9%

H) Between 9% and 11%

I) Between 11% and 15%

J) Bigger than 15%.

Exercise 2 2.1. Write each expression as a single logarithm and, if possible, simplify. ln (x - 4) - ln (x+ 2); ln (x) - 3 [ln (x - 5) + ln (x + 5)] ; log x − 3log(x – 1) 2.2. Solve for x. ln(x – 1)= 1 ; e2x = 4 ; log3x + log3(x2 – 8) = log38x ; 4x2(2x) − 9(2x) = 0

1.Find the monthly payments on a $178,000 15 year mortgage assuming i(2)=6.7%.

2.Find the present value of a perpituity paying $590 at the end of each year assuming interest rates are i(1)=3.6%  and the first payment is in 1 year.

3.Find the present value of an annuity paying $650 per month for 10 years assuming i(4)=14%.

4.

find the maclaurin series for f and its radius of convergence.

(1) f(x) = 10^x
(2) f(x) = e^x^2
(3) f(x) = (1-x)^-5
(4) f(x) = ln(1+x^2)

PLEASE USING THIS INFORMATION FILL OUT ALL COLUMNS COMPLETELY IN THE TABLE MARKED AS (*******)

I only need question C answered, but I provided the rest of the worksheet to use as reference in order to do so.

Q1. Jamie wants to forecast the number of students who will enroll in operations management next semester in order to determine how many sections to schedule. He has accumulated the following enrollment data for the past six semesters:

a (2 pts). Compute a three-semester moving average forecast for semesters 4 through 7 (Model a) (Use two decimals).

b (3 pts). Compute the exponentially smoothed forecast (α = .20) for semesters 1 through 7 (Model b) (Use two decimals).

c (3 pts). Compare the two forecasts using MAD and indicate the most accurate.

MAD (model a) =

MAD (model b) =

We first look at a particle that moves in a one-dimensional potential with form:
? (?) = ?₀ (1/2 ((? / ?) ^ 4) - ((? / ?) ^ 2)),
where ?₀ is a constant with unit Joule and ? a constant with unit meter. We can also imagine
a small sphere influenced by the gravitational acceleration ? that rolls along a roller coaster, where
the height above the ground can be described as:
ℎ (?) = ℎ₀ (1/2 ((? / ?) ^ 4) - ((? / ?) ^ 2)),
so that the potential energy of a mass of mass ? becomes:
? (?) = ??ℎ₀ (1/2 ((? / ?) ^ 4) - ((? / ?) ^ 2))

b) Show that the force acting on a mass of mass ? in position ? is:
? (?) = 2?? (ℎ₀ / ?) * (? / ?− ((? / ?) ^ 3))

c) Find the equilibrium points and characterize them as stable or unstable.

d) You release a particle without initial velocity at a position ?> 0. Where to escape from
that the particle should reach a position with ? <0?

We are now looking at a situation where a particle A with mass ?? = ? is released without initial velocity at
position ?? = −2?. A particle B with mass ?? = 2? is at rest in position ? = −?. The two particles
collide in an inelastic collision with the recovery coefficient
??? = - ((?_A,₁ - ?_B,₁) / (?_A,0 - v_B,0)) = 0.5
where ?_A,0 and ?_A,1 are the velocities of particle A immediately before and after the collision, and correspondingly
for particle B.

e) Find the velocity of particle A just before it collides with particle B at position ? = −?.

f) Find the velocities ?_A,1 and v_B,1 immediately after the collision.

Sugar Land Company is considering adding a new line to its product mix, and the capital budgeting analysis is being conducted by a MBA student. The production line would be set up in unused space in Sugar Land’ main plant. Total cost of the machine is $260,000. The machinery has an economic life of 4 years, and MACRS will be used for depreciation. The machine will have a salvage value of 40,000 after 4 years.


The new line will generate Sales of 1,350 units per year for 4 years and the variable cost per unit is $100 in the first year. Each unit can be sold for $200 in the first year. The sales price and variable cost are expected to increase by 3% per year due to inflation. Further, to handle the new line, the firm’s net working capital would have to increase by $30,000 at time zero (The NWC will be recouped in year 4). The firm’s tax rate is 40% and its weighted average cost of capital is 10%.

1. What are the annual depreciation expenses for years 1 through 4? (10 Points)

2. Calculate the annual sales revenues and costs (other than depreciation), years 1 through 4. (10 points)

3. Estimate annual (Year 1 through 4) operating cash flows (40 points)

4. Estimate the after tax salvage cash flow (10 points)
5. Estimate the cash flow of this project (10 Points)

6. Estimate the NPV, IRR, MIRR, and profitability Index of the project. (20 points)