An investor can design a risky portfolio based on two stocks, A and B. Stock A has an expected return of 15% and a standard deviation of return of 25%. Stock B has an expected return of 12% and a standard deviation of return of 20%. The correlation coefficient between the returns of A and B is 0.2. The risk-free rate of return is 1.5%.

1) Approximately what is the proportion of the optimal risky portfolio that should be invested in Stock B?

2) What is the Expected Return on the Optimal Portfolio?

3) What is the REWARD to VARIABILITY Ratio of the Optimal Portfolio?

A spring mass system with a natural frequency fn = 12 Hz is attached to a vibration table. The table is set to vibrate at 16 Hz, 0.5 g maximum acceleration:

a. What is the amplitude of the table's motion in inches?

b. What is the magnification factor M of this undamped system?

c. What is the maximum displacement of the mass assuming no dampening?

d. What is the maximum acceleration of the mass assuming the packaging can be modeled as a viscous damper with a damping ratio of 0.2?

e. Is the motion of the mass in phase or out of phase with the motion of the table?

The payoff table below provides the profits (in thousands of dollars) for each of four alternatives in each of three supplies.

Suppose that the probabilities for the supplies above are P(S₁) = 0.6, P(S₂) = 0.2, and P(S₃) = 0.1.

1. Which alternative should be selected under Bayes’ Rule?
2. What is the expected value of perfect information for this decision?

A company that produces computers claims that that the average life of one of their computers is 8 years. A sample of 12 computers shows a sample mean life of 7.8 years, with a population standard deviation of 0.2 years. Does the data suggest that the average life of one of the computers is not 8 years at 0.05 level significance? Assume computer lifetimes are normally distributed.

1. Give null and alternate hypothesis

2. Give test statistic and P value, state conclusion

3. State what type 1 and type 2 errors would be

Consider the following time series data.

a. develop the three-week moving average forecasts for this time series. Compute MSE and a forecast for week 7 (to 2 decimals if necessary).

b. use = 0.2 to compute the exponential smoothing forecasts for the time series. Compute MSE and a forecast for week 7 (to 2 decimals).

c. use a smoothing constant of =0.4 to compute the MSE (to 2 decimals).

A red blood cell has a disk diameter of 0.6 micrometers and an intracellular isotonic osmolarity of 290mOsmol/l at a temperature of 293 kelvin. The hydraulic permeability of the red blood cell is 2.4 micrometer atm^-1min^-1 with a b inactive fraction intercept of 0.2. Calculate the normalized cell volume and plot it against osmolarity variations between 50mOsm to 850mOsm. Plot another graph that represents cell volume variations against time when the extracellular isotonic osmolarity varies from 290mOsmol/l to 990mOsm/l.

A) Solve the initial value problem:

8x−4y√(x^2+1) * dy/dx=0

y(0)=−8

y(x)=

B)  Find the function y=y(x) (for x>0 ) which satisfies the separable differential equation

dy/dx=(10+16x)/xy^2 ; x>0

with the initial condition y(1)=2
y=

C) Find the solution to the differential equation

dy/dt=0.2(y−150)

if y=30 when t=0

y=

Determine the quality (if saturated) or temperature (if superheated) of the following: (show details of your answer, not just write the final answer)

a) Water at p=50 kPa, v=1 m3/kg

b) Water at p=1.6 MPa, v=0.15 m3/kg

c) Water at p=7 MPa, h=3160 kJ/kg

d) Refrigerant R-134a at p=200 kPa, v=0.009 m3/kg

e) Refrigerant R-134a at p=0.2 MPa, v=0.12 m3/kg

1. Consider water flowing upwards through the 22.5^o reducing elbow (22.5^o angle to horizontal axis). If diameters of the elbow at entrance point and exit points are 0.02540 m and 0.01905 m respectively, elbow is 0.1 m high, velocities of water are 5 m/s and 9 m/s at entrance point and exit point respectively, and density of water is 1000 kg/m³_, moment flux correction factor = 1.02.
2. 1. Determine gage pressure at inlet
3. 2. Determine vertical component of force of the system, if weight of the system is 0.2 kg.

1. The length of a magnetic circuit of a relay is 25 cm and the cross-sectional area is 6.25 cm². The length of the air-gap in the operated position of the relay is 0.2 mm. Calculate the magnetomotive force required to produce a flux of 1.25 mWb in the air gap. The relative permeability of magnetic material at this flux density is 200. Calculate also the reluctance of the magnetic circuit when the relay is in the unoperated position, the air-gap then being 8 mm long (assume μ_r remains constant).

[2307 AT, 1.18 x 10⁷ AT/Wb]