Let X denote a random variable that follows a binomial distribution with parameters n=5, p=0.3, and Y denote a random variable that has a Poisson distribution with parameter λ = 6. Additionally, assume that X and Y are independent random variables.

1. What are the possible values for (X, Y ) pairs.

2. Derive the joint probability distribution function for X and Y. Make sure to explain your steps.

3. Using the joint pdf function of X and Y, form the summation /integration (whichever is relevant) that gives the expected value for X4 + Y + 7.

4. Using the joint pdf function of X and Y, set up the summation /integration (whichever is relevant) that gives the expected value for X, and COMPUTE its value.

Let X denote a random variable that follows a binomial distribution with parameters n=5, p=0.3, and Y denote a random variable that has a Poisson distribution with parameter λ = 6. Additionally, assume that X and Y are independent random variables.

1. What are the possible values for (X, Y ) pairs.

2. Derive the joint probability distribution function for X and Y. Make sure to explain your steps.

3. Using the joint pdf function of X and Y, form the summation /integration (whichever is relevant) that gives the expected value for X^4 + Y + 7.

4. Using the joint pdf function of X and Y, set up the summation /integration (whichever is relevant) that gives the expected value for X, and COMPUTE its value.

Consider the following scenario analysis:

Assume a portfolio with weights of 0.60 in stocks and 0.40 in bonds.

1. What is the rate of return on the portfolio in each scenario? (Enter your answer as a percent rounded to 1 decimal place.)

b. What are the expected rate of return and standard deviation of the portfolio? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

c. Would you prefer to invest in the portfolio, in stocks only, or in bonds only? Explain the benefit of diversification.

Use 5 months moving average and exponential smoothing method (using alpha=0.3) on the following sale of Maggi. You are supposed to forecast the sale of Maggi in the 13^th Month. Compare whether there is any difference in forecast based on two methods and Comment.

A pitched baseball (m=0.3 kg) reaches a catcher’s glove travelling at a velocity of 28 m/s. Calculate the ball’s: A) momentum, B) kinetic energy. How much impulse is required to catch the ball? If the hand moves backward 32cm while the person is catching it, what is the average force applied to the ball? How much time will it take before the ball stops? What is the average power of this contact?

The variation of the spectral transmissivity of a 0.6cm thick glass window is 0.92 in the wavelength range from 0.3-3micrometer. Determine the average transmissvity of this window for solar radiation at T=5800K. Also, determine the amount of solar radiation transmitted through the window for incident solar radiation of 650W/m².

2. Let X be a continuous random variable with PDF ?fx(x)= cx(1 − x), 0 < x < 1,

0 elsewhere.

(a) Find the value of c such that fX(x) is indeed a PDF.

(b) Find P(−0.5 < X < 0.3).

(c) Find the median of X.

describe how to make one liter of the following solution: 0.3 M mannitol (FW=182), 0.02 M KPO4, 0.01 M KCl, 0.005 M MgCl2. You have the following stocks... 1 M KPO4, 1 M KCl, 1 M MgCl2.

A capacitor is connected to a battery and placed in a magnetic field (-z direction) to form a velocity selector. A charge moving at 3,479.6 m/s is not deflected by the velocity selector. If the voltage of the battery is 7 volts and the distance between the sheets is 0.3 meters, what is the amount of the magnetic field in milli-Tesla?

Shaving Cream

You are working with the marketing team for a FMCG firm that produces shaving cream. The team believes that sales of some of the products are closely related to sales of other products. They want you to explore this in a little more depth for two products, SKU 123 and SKU 456. Unfortunately, all of the base sales data for these products has been destroyed. All that you have is the weekly summary data:

The marketing team believes the correlation of these items is 0.8.

Question 1 What would the covariance need to be for the marketing team to be correct?

Now the marketing team wants to understand the potential weekly sales for these two products. Let the sales price for the two SKUs be 12.50, 7.75, respectively.

Question 2 What is the expected weekly revenue?

Assume that marketing is correct and the correlation = 0.8.

Question 3 What is the standard deviation of the weekly revenue?

Question 4 Assuming the joint distribution is normal, and the marketing team’s correlation of 0.8 is correct. What is the probability that weekly sales will be between 10,000 and 20,000 dollars?

Question 5

Which of the following statements are true:

Choose the correct answer.

A: The population mean is always greater than the sample mean.

B: Everything else being equal, the Confidence Interval for a sample increases with the number of observations (n) in the sample.

C: The t-distribution is used for small sample sizes instead of the Normal.

D: Holding all else equal, reducing the probability of a Type I error actually increases the probability of a Type II error.