A fair die is rolled, and then a coin with probability p of Heads is flipped as many times

as the die roll says, e.g., if the result of the die roll is a 3, then the coin is flipped 3

times. Let X be the result of the die roll and Y be the number of times the coin lands

Heads.

(a) Find the joint PMF of X and Y . Are they independent?

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

(c) Find the conditional PMFs of Y given X = x and of Y given X = x.

6. A committee of size k is chosen from a group of n women and m men. All possible

committees of size k are equally likely. Let X and Y be the numbers of women and men

on the committee, respectively.

(a) Find the joint PMF of X and Y . Be sure to specify the support.

(b) Find the marginal PMF of X in two di?erent ways: by doing a computation using

the joint PMF, and using a story.

(c) Find the conditional PMF of Y given that X = x.

7. A stick of length L (a positive constant) is broken at a uniformly random point X.

Given that X = x, another breakpoint Y is chosen uniformly on the interval [0, x].

(a) Find the joint PDF of X and Y . Be sure to specify the support.

(b) We already know that the marginal distribution of X is Unif(0, L). Check that

marginalizing out Y from the joint PDF agrees that this is the marginal distribution of

X.

(c) We already know that the conditional distribution of Y given X = x is Unif(0, x).

Check that using the definition of conditional PDFs (in terms of joint and marginal

PDFs) agrees that this is the conditional distribution of Y given X = x.

(d) Find the marginal PDF of Y .

(e) Find the conditional PDF of X given Y = y.

8. (a) Five cards are randomly chosen from a standard deck, one at a time with replacement.

Let X, Y, Z be the numbers of chosen queens, kings, and other cards. Find the joint PMF

of X, Y, Z.

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

Hint: In summing the joint PMF of X, Y, Z over the possible values of Z, note that most

terms are 0 because of the constraint that the number of chosen cards is five.

(c) Now assume instead that the sampling is without replacement (all 5-card hands are

equally likely). Find the joint PMF of X, Y, Z.

Hint: Use the naive definition of probability.

9. Let X and Y be i.i.d. Geom(p), and N = X + Y .

(a) Find the joint PMF of X, Y, N.

(b) Find the joint PMF of X and N.

(c) Find the conditional PMF of X given N = n, and give a simple description in words

of what the result says

A report states that the mean yearly salary offer for students graduating with mathematics and statistics degrees is $62,915. Suppose that a random sample of 50 mathematics and statistics graduates at a large university who received job offers resulted in a mean offer of $63,600 and a standard deviation of $3,900. Do the sample data provide strong support for the claim that the mean salary offer for mathematics and statistics graduates of this university is greater than the national average of $62,915? Test the relevant hypotheses using a = 0.05. Find the test statistic and P-value. (Use technology to calculate the P-value. Round your test statistic to two decimal places and your P-value to three decimal places.) t= P-value = State your conclusion. (A) Reject HO. We have convincing evidence that the mean salary offer for mathematics and statistics graduates of this university is greater than the national average of $62,915. (B) Fail to reject HO. We have convincing evidence that the mean salary offer for mathematics and statistics graduates of this university is greater than the national average of $62,915. (C) Reject HO. We do not have convincing evidence that the mean salary offer for mathematics and statistics graduates of this university is greater than the national average of $62,915. (D) Fail to reject HO. We do not have convincing evidence that the mean salary offer for mathematics and statistics graduates of this university is greater than the national average of $62,915.O 70 80 90 100 The second box-and-whisker plot is shifted to the right and less spread out (10% less). Lower extreme = 81,lower Q = 64,median = 74,upper Q = 85,upper extreme = 94 O 60 70 80 90 The second box-and-whisker plot is shifted to the right and less spread out (10% less). Lower extreme = 64,lower Q = 74,median = 81,upper Q = 85, upper extreme = 94 O 80 90 100 The second box-and-whisker plot is shifted to the right and more spread out (10% more). Lower extreme = 64,lower Q = 74,median = 81,upper Q = 85, upper extreme = 94 O 60 70 90 The second box-and-whisker plot is shifted to the right and less spread out (10% less). Lower extreme = 85,lower Q = 74,median = 81,upper Q = 64,upper extreme = 94 O 70 90 100 The second box-and-whisker plot is shifted to the right and more spread out (10% more).Campbell Manufacturing Company began operations on January 1. During the year, it started and completed 1,650 units of product. The financial statements are prepared in accordance with GAAP. The company incurred the following costs: 1. Raw materials purchased and used---$3,150. 2.Wages of production workers--$3,490. 3.Salaries of administrative and sales personnel--$1,960. 4.Depreciation of manufacturing equipment--$4,415. 5. Depreciation on administrative equipment--$1,830. Campbell sold 1,130 units of product. Required a. Determine the total product cost for the year. b. Determine the total cost of the ending inventory. (Do not round intermediate calculations.) c. Determine the total of cost of goods sold. (Do not round intermediate calculations.)