Suppose that T (≥ 0) is a continuous random variable. Let its pdf, cdf, survival function, hazard rate function, and cumulative hazard rate function be f(t), F(t), s(t), h(t), and H(t), respectively. Note that H(t) is defined by R t 0 h(u)du.

a. Denote s(t), h(t), and H(t) as a function of f(t).

b. Denote f(t), h(t), and H(t) as a function of s(t).

c. Denote f(t), s(t), and H(t) as a function of h(t).

d. Denote f(t) and s(t) as a function of H(t).

1. Suppose that X is a Geometric r.v. and its pmf is

f(x; p) = p(1 − p) x−1 , x = 1, 2, . . . and 0 ≤ p ≤ 1.

a. Calculate the mean of X.

b. Calculate the median of X.

c. Check and show whether the median is smaller than mean.

A burger place offers its burgers with the choice of one, two, or three meat patties. Customers may add any (as many or none as they choose)

of the following condiments: ketchup, mustard, tomato, lettuce, pickles, mayonnaise, cheese, and onions. How many different kinds of hamburgers can be ordered?

1. Let’s assume you have a bag of balls (5 red and 5 black). You roll a six-sided dice and then select a ball from the bag.
   1. What is the probability you roll a five on the die and then select a red ball?
2. You are playing blackjack with no other players (i.e. cards are dealt immediately with no one else receiving cards). What is the probability you get a K, Q, J, or 10 on the first card followed by an ace on the second card? (No replacement)
3. The Powerball Lottery requires you to select 5 numbers between 1 and 69 and 1 number between 1 and 26. What is the probability of selecting ALL numbers correct? (Please show your work).

1.) Based on your answers on part a and d, are sex and college major of students in this class independent? Provide a mathematical argument? (Problem A asked probability students are male which was 30/53=0.57 and Problem D asked probability student is industrial engineer given the student is male which ended up being 0.60)

2.) Consider the events A and B. Are sex and college major mutually exclusive events? Provide a mathematical argument to justify your answer.

3.) Find the probability that the randomly selected student is male or industrial engineering college major.

4.) Consider the events C and D. Are college major mutually exclusive events? Provide a mathematical argument to justify your answer.

Four numbers are selected without replacement from {1,2,3,4,5,6,7} to form a 4-digit number. What is the probability that the number is greater than 6543?

Given two independent random samples with the following results: n1=15x‾1=125s1=26 n 1 =15 x ‾ 1 =125 s 1 =26    n2=11x‾2=159s2=23 n 2 =11 x ‾ 2 =159 s 2 =23 Use this data to find the 98% confidence interval for the true difference between the population means. Assume that the population variances are equal and that the two populations are normally distributed.

Step 2 of 3 : Find the standard error of the sampling distribution to be used in constructing the confidence interval. Round your answer to the nearest whole number.

Step 3 of 3: Find the lower and upper limits of the confidence interval

Alfraed and Broetschen have each put $500 into a pool and bet on the winner of the 2057 Stanley cup which goes to the first team to win 4 games out of seven matches. All other teams having been disbanded due to dental injuries, the contest is between the Waterloo Weirdos and the Kitchener Klutzes and after one game, Kitchener is ahead one game to zero in the best of 7 final. Alfraed, who cleverly bet on Kitchener, wants out of the bet, since he needs money immediately to cover a poker debt to a sadistic biker in Accounting.

a. Assuming the teams are evenly matched, what is the probability that Alfraed would win the total stake ($ 1000) if he continued?

b. What would be a fair division of the stakes (the total amount at stake is $1000) if the bet is terminated? By how much does Alfraed profit from the bet in this case?

A grocery bag can be classified as either paper or plastic . Suppose that 92 ​% of grocery bags are classified as plastic . ​(a) Two grocery bags are chosen at random. What is the probability that both grocery bags are plastic ​? ​(b) Nine grocery bags are chosen at random. What is the probability that all nine grocery bags are plastic ​? ​(c) What is the probability that at least one of nine randomly selected grocery bags is paper ​? Would it be unusual that at least one of nine randomly selected grocery bags is paper ​?

Let X and Y be random variables with means µX and µY . The covariance of X and Y is given by, Cov(X, Y ) = E[(X − µX)(Y − µY )]

a) Prove the following three equalities: Cov(X, Y ) = E[(X − µX)Y ] = E[X(Y − µY )] = E(XY ) − µXµY

b) Suppose that E(Y |X) = E(Y ). Show that Cov(X, Y ) = 0 (hint: use the law of interated expectations to show that E(XY ) = µXµY ). In this case, what is the correlation coefficient ρ between X and Y , equal to?

c) Suppose that Cov(X, Y ) = 0. Does this imply that X and Y are independent? Explain your reasoning.

What logical or ethical problems do you see in these hypothetical scenarios?

a. Dolon Privacy Consultants concludes that its employees are not loyal because a few samples of employee e-mails contained comments critical of the company’s management.

b. Calchas Financial Advisors issues a glowing report of its new stock market forecasting system, based on testimonials of five happy customers.

c. Five sanitation crew members at Malcheon Hospital are asked to try a new cleaning solvent to see if it has any allergic or other harmful side effects.

d. A consumer group rates a new personal watercraft from Thetis Aquatic Conveyances as “Unacceptable” because two Ohio teens lost control and crashed into a dock.

Consider the following observations on shear strength (MPa) of a joint bonded in a particular manner.

c)How large or small does an observation have to be to qualify as an outlier? (Round your answers to one decimal place.) above and below

How large or small does an observation have to be to qualify as an extreme outlier? (Round your answers to one decimal place.) above and below

d)By how much could the largest observation be decreased without affecting f_s?

Poly(3-hydroxybutyrate) (PHB), a semicrystalline polymer that is fully biodegradable and biocompatible, is obtained from renewable resources. From a sustainability perspective, PHB offers many attractive properties though it is more expensive to produce than standard plastics. The accompanying data on melting point (°C) for each of 12 specimens of the polymer using a differential scanning calorimeter appeared in an article.

b) the sample variance s² from the definition [Hint: First subtract 180 from each observation.]

c)the sample standard deviation

d)s² using the shortcut method

Consider the data in the file Growth.csvwhich contains data on average growth rates over 1960-1995 for 65 countries, along with variables that are potentially related to growth. A complete description of the data is given in data description which is under the name Growth- Data Description and can be found on Blackboard.

Using this data, carry out the following empirical exercises:

Construct a table that shows the sample mean, std. deviation, minimum and maximum values for the variablesGrowth, Trade-Share, YearsSchool, Oil, Rev_Coups, Assasinations, and RGDP60.

Run a regression of Growthon TradeShare, YearsSchool,Rec_Coups, Assasinations and RGDP60.Show the output of this regression in R (take a screenshot and crop the output summary in your word file that contains your answers).

What is the value on the coefficient on Rev_Coups?Is it statistically significant? Do interpretation on this coefficient.

What is the value of the adjusted R-square? Do these variables explain the majority of country growth?

Use the regression to predict the average annual growth for a country that has average values for all regressors.

Repeat (3) but now assume that the country’s value for TradeShareis one std. deviation above its mean.

Notes: Please submit through email an individual word file with your answers WELL ORGANIZEDalong with your R code (.R file) at the end.