Probability and Statistics

Write a Matlab simulation to solve problem 2.4.4.

2.4.4.

Each problem has a failure probability q, independent of any other component. A successful operation requires:

1. components 1, 2, and 3 all work, or 4 works

2. component 5 or component 6 works.

with q = 0.2, simulate the replacement of a component with an ultrareliable component. For each replacement of a regular component, perform 100 trials.

Assume that you have half your money invested in Times Mirror, the media company, and the other half invested in Unilever, the consumer product giant. The expected returns and standard deviations on the two investments are summarized below:

----------------------Times Mirror------ Unilever

Expected Return 14%------------- 18%

Standard Deviation 25% ------------40%

Estimate the variance of the portfolio as a function of the correlation coefficient (Start with –1 and increase the correlation to +1 in 0.2 increments).

Based on the following data;

State of the Economy               Probability      Stock A Rate of Return   Stock B Rate of Return

Recession                                      0.2                                4%                                  -20%

Normal Growth                            0.65                               8%                                   20%

Boom                                             0.15                             16%                                   60%   

(a) calculate the expected return and the standard deviation of returns for each stock.

(b) Calculate the expected return and the standard deviation on the portfolio, where

the portfolio is formed by investing 50% of the funds in Stock A and the rest in Stock B.

1) Suppose 500 mL of an 0.200 molar HCl acid is titrated with 70.0 mL of a 1.00 M NaOH solution. What is the final pH of the solution? The error interval is 0.1 pH units.

2) Suppose 500 mL of 0.100 M benzoic acid, which has a pK_a of 4.201, solution is titrated with 75.0 mL of a 1.00 M NaOH. What is the final pH? The error interval is +/- 0.2 pH units.

You mix together two flows of water. The first flow is a saturated liquid at 300 kPa, moving at a rate of 2 kg/s. The second flow is a superheated vapor at 300 kPa and 300 °C, and is moving at a rate of 0.2 kg/s. If the outlet is similarly at 300 kPa, (a) what is the mass flow rate of the outgoing water, (b) what is the temperature of the outgoing water, and (c) What is the rate of entropy generation for this process?

Bags of potato chips have a mean weight of 6 ounces with a standard deviation of 0.2 ounces. There are 100 bags of potato chips in a box.

(i) What is the probability that the total weight of the 100 bags is greater than 603 ounces?

(ii) A potato chip factory produces 1000 boxes of potato chips. What is the probability that more than 70 of these boxes contain more than 603 ounces of potato chips?

A charge of -0.4 µC is located at the origin; a charge of 0.53 µC is located at x = 0.2 m, y = 0; a third charge Q is located at x = 0.32 m, y = 0. The force on the 0.53 µC charge is 4.3 N, directed in the positive x direction.

1. Determine the charge

2. With this configuration of three charges, where, along the x direction, is the electric field zero? xf=

3. x2=

A college senior must choose between two Choices: going for an MBA or taking a full-time entry-level-level position right after graduation. She thinks that she has 0.6 probability of completing the MBA in a year. If she completes the MBA, she believes that she has 0.1 probability of getting a manager position; otherwise, she will get a senior staff position. Should she fails the MBA, she will have to take the entry job but with less seniority than what she would have if she had gone to work right after graduation. Once started at the entry-level position for a year, she believes that she has a 50-50 chance of moving up to a junior staff position versus staying at the entry-level position. Her preferences for the possible outcomes of her choice at the end of two years are listed in decreasing order below:

(1) Completing the MBA and getting a management position

(2) Completing the MBA and getting a senior staff position

(3) Moving to junior staff without going to MBA and thus more seniority

(4) Moving to junior staff after failing the MBA

(5) Staying at entry level without going to MBA and thus more seniority

(6) Staying at entry level after failing the MBA

Using the simple decision tree for utility estimation, she has found that she would be indifferent between:

Outcome (2) and a lottery with a 50-50 chance of yielding the best outcome (1) and the worst outcome (6)

Outcome (3) and the lottery if the lottery has a 0.35 probability yielding (1) and a 0.65 probability of yielding (6).

Outcome (4) and the lottery if the lottery has a 0.2 probability yielding (1) and a 0.8 probability of yielding (6).

Outcome (5) and the lottery if the lottery has a 0.1 probability yielding (1) and a 0.9 probability of yielding (6).

6a (10 points) By assigning a utility 0 to (6) and 100 to (1), find the utility for each of the four outcomes between (1) and (6).

6b (10 points) Draw a decision tree for her career decision and find her best Choice for the two-year period

Two rural individuals Matyás and Toka are contemplating leaving their village for a nearby town. Given that Matyás has a secondary school certificate, she has a higher probability of obtaining an urban sector job than Toka, who is a primary school drop out. If they remain in the village and work on their family farms, they earn $200 each per year if the yield is high and $100 per year if the yield is low. The yield will be high with probability 0.5 and low with probability 0.5. Alternatively both these individuals can choose to migrate to town, in which case they each expect to find a formal or informal sector job, paying the following earnings according to the given probabilities.

1. (a) According to the Harris-Todaro model, on a one-year time horizon, which individual will migrate? [2 points]

2. (b) On the three-year time horizon, which individual(s) will migrate? (Assume the rate of interest at which both can borrow or lend money is i = 5%) [5 points]

3. (c) Assume a one-year time horizon. Suppose now that there is a traditional family support payment of $40 that all urban migrants must pay yearly to their relatives back in the village. At the same time, each bears a ‘psychological cost’ associated with moving to the city. Such a cost is $25 for Matyás and $15 for Toka. Again, according to the Harris-Todaro model, which farmer will migrate to town? [5 points]

4. (d) Assume a one-year time horizon, Still maintaining the ‘psychological’ costs associated with moving to the city, suppose now that all formal jobs also come with medical benefits, which are worth $30 a year, in additional to the wage. Informal sector jobs do not carry any such benefits. According to the Harris- Todaro model once again, which individuals will migrate to town? [3 points]