A 95% confidence interval for the unknown expectation of some

distribution contains the number 0.

a. We construct the corresponding 98% confidence interval, using the same

data. Will it contain the number 0?

b. The confidence interval in fact is a bootstrap confidence interval. We repeat the bootstrap experiment (using the same data) and construct a new

95% confidence interval based on the results. Will it contain the number 0?

c. We collect new data, resulting in a dataset of the same size. With this data,

we construct a 95% confidence interval for the unknown expectation. Will

the interval contain 0?

Alice and Bob arrange to meet for lunch on a certain day at noon. However, neither is

known for punctuality. They both arrive independently at uniformly distributed times

between noon and 1 pm on that day. Each is willing to wait up to 15 minutes for the

other to show up. What is the probability they will meet for lunch that day?

2. Alice, Bob, and Carl arrange to meet for lunch on a certain day. They arrive independently at uniformly distributed times between 1 pm and 1:30 pm on that day.

(a) What is the probability that Carl arrives first?

For the rest of this problem, assume that Carl arrives first at 1:10 pm, and condition on

this fact.

(b) What is the probability that Carl will have to wait more than 10 minutes for one of

the others to show up? (So consider Carl's waiting time until at least one of the others

has arrived.)

(c) What is the probability that Carl will have to wait more than 10 minutes for both

of the others to show up? (So consider Carl's waiting time until both of the others has

arrived.)

(d) What is the probability that the person who arrives second will have to wait more

than 5 minutes for the third person to show up?

3. One of two doctors, Dr. Hibbert and Dr. Nick, is called upon to perform a series of

n surgeries. Let H be the indicator r.v. for Dr. Hibbert performing the surgeries, and

suppose that E(H) = p. Given that Dr. Hibbert is performing the surgeries, each surgery

is successful with probability a, independently. Given that Dr. Nick is performing the

surgeries, each surgery is successful with probability b, independently. Let X be the

number of successful surgeries.

(a) Find the joint PMF of H and X.

(b) Find the marginal PMF of X.

(c) Find the conditional PMF of H given X = k.

4. A fair coin is flipped twice. Let X be the number of Heads in the two tosses, and Y be

the indicator r.v for the tosses landing the same way.

(a) Find the joint PMF of X and Y .

(b) Find the marginal PMFs of X and Y .

(c) Are X and Y independent?

(d) Find the conditional PMFs of Y given X = x and of X given Y = y

Events Ay and A, are mutually exclusive and form a complete partition of a sample space $ with P(A, ) = 0.75 . and P(A,) = 0.25. If event E is an event in S with P(E) A, ) = 0.035 and P(E | A, ) = 0.106 ,using Bayes theorem compute P(A |E) -? (Round your final answer to four decimal places). 0 04608 0 0.5723 0 0.3946 O 0.4976 Q:0.5236 0 0.6152 O 0.3822 Correct answer is not listed14. Average grade in business statistics course is 82 in 2019. In addition the standard deviation is 8. If we assume that the distribution of grades is normally distributed, calculated following probabilities (7 points). (numbers are arbitrary). a. probability of a grade less than 90. b. probability of a student's grade greater than 75. c. probability of a grade between 70-803. You have a coin which is 'heads' with probability - and 'tails' with probability . Suppose you flip your coin twice (the two flips are independent). (i) Let A be the event that the first flip is 'heads' and the second flip is 'tails'. What is Pr(A)? [1] (ii) Let B be the event that the two flips are different. What is Pr( B | A)? [2] (iii) Use Bayes' theorem to calculate Pr(A | B) (no marks without using Bayes' theorem). [3]