2 Consider the probability density function (p.d.f) of a continuous random variable X: f(x) = ( k x3 , 0 < x < 1, 0, elsewhere, where k is a constant. (a) Find k. (b) Compute the cumulative distribution function F(x) of X. (c) Evaluate P(0.1 < X < 0.8). (d) Compute µX = E(X) and σX.

An audio amplfier drives a series RLC circuit consisting of an 8.0 Ω loudspeaker, a C = 42 × 10−10F capacitor, and a L = 60 × 10−6 H inductor. The amplifier output is voltage is 15.0 V at a frequency of f = 106 Hz. The current flowing in the circuit is given by

(a) 0.044 A (b) 1.875 A (c) 0.8 A (d) 12 A (e) None of the above.

1a. A portfolio manager adds a new stock that has the same standard deviation of return as the existing portfolio but has a correlation coefficient with the existing portfolio that is less than +1. Explain why the standard deviation of the portfolio will decrease?

1b. A portfolio has assets with coefficient of correlation of 0.8. show whether sharpe ratio or Treynor ratio is appropriate for measuring the risk-adjusted return?

In the last quarter of​ 2007, a group of 64 mutual funds had a mean return of 0.8​% with a standard deviation of 4.9​%. Consider the Normal model ​N(0.008​,0.049​) for the returns of these mutual funds.

​a) What value represents the 40th percentile of these​ returns?

​b) What value represents the 99th​ percentile?

​c) What's the​ IQR, or interquartile​ range, of the quarterly returns for this group of​ funds?

In a city of 80,000 households, 800 were randomly sampled in a marketing survey by an auto manufacturer. To see how typical this city was, some statistics were collected: the average number of cars per household was 1.32, while the standard deviation for the number of cars per household was 0.8. Construct a 95% confidence interval for the average number of cars per household in the whole city.

Suppose that all the numbers in an array are located in an interval [0,12], and we need to find the largest element with accuracy ε = 0.8. How many iterations will you need if we use the quantum optimization algorithm? How many times do we need to apply Grover's algorithm? Trace the quantum optimization algorithm for the case when the actual largest element is a5 = 3.14

Steam issues from the nozzle of a De-level turbine at a velocity of 1000 m/s at an angle of 20ᵒ. The
blade velocity is 300 m/s and blades are symmetrical. The mass flow rate is 0.5 kg/s and a friction
factor is 0.8. With the help of velocity diagram, determine 1) Blade efficiency 2) Power developed 3)
Stage efficiency if the nozzle efficiency is 95%

.   If you have $30,000 invested in each of two stocks and $20,000 invested in each of another three stocks and the betas on the stocks above are 0.8, 1.1, 1.0, 1.2 and 1.4, respectively, what is the beta of your portfolio, and what is the required return on the portfolio if the risk-free rate is 4.6% and the return on the market portfolio is 10.4%? What are the risk premiums for the market and for your portfolio?