1) An urn contains 10 balls, 2 red, 5 blue, and 3 green balls. Take out 3 balls at a random, without replacement. You win $2 for each green ball you select and lose $3 for each red ball you select. Let the random variable X denote the amount you win, determine the probability mass function of X.

2) Each of the 60 students in a class belongs to exactly one of the three groups A,B, or C. The membership numbers for the 3 groups are as follows:

A: 15 students

B: 20 C: 25 First, Next,

a. b. c.

students
students
choose one of the 60 students at random and let X be the size of that student’s group;.

choose one of the 3 groups at random and let Y be its size. Write down the pmf for X and Y.
Compute ?(?) ??? ?(?).
Compute ???(?) ??? ???(?).

3. Roll a fair die repeatedly. Let X denote the number of 4’s in the first 6 rolls and let Y denote the number of rolls needed to obtain a 2.

a. WritedowntheprobabilitymassfunctionofX. b. WritedowntheprobabilitymassfunctionofY. c. Find an expression for ?(? ≥ 3).
d. Find an expression for ?(? > 6).

1. A study of sterility in the fruit fly reports the following data on the number of ovaries developed by each female fly in a sample of size 1386. One model for unilateral sterility states that each ovary develops with some probability p independently of the other ovary. Test the fit of this model using χ². (Use α = 0.05.)

Calculate the test statistic (Round your answer to two decimal places

2. The accompanying data refers to leaf marks found on white clover samples selected from both long-grass areas and short-grass areas. Use a χ² test to decide whether the true proportions of different marks are identical for the two types of regions. (Use α = 0.01.)

Type of Mark

Calculate the test statistic (Round your answer to two decimal places)

Question 1:

1. The following table gives the frequency distribution of the number of orders received each day during the past 50 days at the office of the mail order company. Calculate the mean, mode, median, quartiles, variance and standard deviation.

b. "Fits", a designer dress retailer specialising in ladies' formal wear, is currently in the process of re-ordering a batch of formal black dinner dresses. From its records of the last 100 sales, the following statistics on the dress sizes sold were calculated:

Mean(x) = 8.75 Md= 7.5 Mo = 8

Which measure of central tendency best describes the average size of dresses sold?

c. Suppose the probability that a house of a certain type will burn down in any 12-month period is 0.004. An insurance company offers to sell the owner of such a house a $120, 000 one-year term fire-insurance policy for a premium of $690. What is the company's expected gain from such a contract?

Subject: Neural network and Pattern recognition (Deep Learning)

Given the following partial network definitioon:

Input - A 28 x 28 RGB image

First Layer - 7, 5 x 5 filters. Activation function: Sigmoid

Second Layer - 23, 3 x 3 filters. Activation function: RelU

Third Layer - 20, 3 x 3 filters. Activation function: RelU

You task is to do design(s) of the next part of the network (so that it uses the output of the third layer).

The purpose of the complete network will be to predict a probability distribution over 5 classes.

You are required to do two designs.

- In the first design you can use fully connected layers.

- In the second design you are cannot use fully connected layers but instead must use convolutional layers.

Your design definition should include

- Any operation you are performing (e.g. flattening, reshaping etc.)

- Any activation function you want to use

- The number of neurons (for fully connected layers)

- The number of filters, their kernel size, and response shape (for convolutional layers)

It is your responsibility, as the new head of the automotive section of Nichols Department Store, to ensure that reorder quantities for the various items have been correctly established. You decide to test one item and choose Michelin tires, XW size 185 × 14 BSW. A perpetual inventory system has been used, so you examine this as well as other records and come up with the following data:

Because customers generally do not wait for tires but go elsewhere, you decide on a service probability of 98 percent. Assume the demand occurs 365 days per year.

a. Determine the order quantity. (Round your answer to the nearest whole number.)

b. Determine the reorder point. (Use Excel's NORM.S.INV() function to find the z value. Round z value to 2 decimal places and final answer to the nearest whole number.)

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation.

(a) What is P(X = 1 and Y = 1)?
P(X = 1 and Y = 1) = _____

(b) Compute P(X ≤ 1 and Y ≤ 1).
P(X ≤ 1 and Y ≤ 1) = ______

Compute the probability of this event.
P(X ≠ 0 and Y ≠ 0) =  

(d) Compute the marginal pmf of X.

Compute the marginal pmf of Y.

Using p_X(x), what is P(X ≤ 1)?
P(X ≤ 1) = ________

Assume Julian has a choice between two deposit accounts. Account A has an annual percentage rate of 7.55 percent but with interest compounded quarterly. Account B has an annual percentage rate of 7.50 percent with interest compounded daily. Which account provides the highest effective annual return?

Account A

Account B

Both provide the same effective annual return.

We don't have sufficient information to make a choice.

Use the following table to answer the question below.

What is the highest fee that a client who is currently borrowing would be willing to pay to invest in ABC fund instead of S&P500?

Round your answer to 4 decimal places. For example, if your answer is 3.205%, then please write down 0.0321.

Project A has a total present value of future cash flows of $98,200 and a cost of $102,000. Project B has a total present value of future cash flows of $386,911 and a cost of $287,211. Project C has a total present value of future cash flows of $784,000 and a cost of $668,000. Project D has a total present value of future cash flows of $119,865 and a cost of $98,770. Which of these projects has the highest Profitability Index?