PLEASE PICK ANY 3 PROBLEMS YOU ARE ABLE TO DO AND LET ME KNOW WHICH ONES YOU WANNA DO. THE PROJECT NEEDS TO BE FINISHED BY FRIDAY NIGHT, NO EXTENSION POSSIBLE.

PLEASE BE HONEST WHEN BIDDING. I WILL SELECT A WINNER EARLY TOMORROW.

1) Toss a fair coin until you get HHH or TTT, but not more than 8

tosses. Let X be the number of tosses and Y = |the number of heads

— the number of tails|. Find the joint distribution of X and Y.

Also compute the correlation coefficient of X and Y.

2) Do 200 experiments of Problem 1, compute the sample mean, sample variance

of X and Y. Also compute the sample covariance and sample

correlation coefficient of X and Y. You have to state how you do

your experiments clearly and provide the raw data.

3) Let U, V, W, X, Y, Z be six independent and identically distributed

geometric random variables with parameter p, where 0 < p < 1. Let

S = max{U,V,W,X,Y,Z}. Find the p.m.f. of S and also find E(S).

(There is plenty of time for evolution)

4) Let U, V, X, Y, Z be five independent and identically distributed

exponential random variables with parameter c, where 0 < c. Let W

= medium of U,V,X,Y,Z. Find the pdf of W and also find E(W) and

Var(W).

5) 3 players A, B, and C take turns at a game according to the

following rules. At the start A and B play while C is out. The

loser is replaced by C and at the second trial the winner plays

against C while the loser is out. The game continues in this way

until a player wins twice in succession, thus becoming the winner

of the game. What is the probability that Player A will be the

winner? What is the probability that Player C will be the winner?

(assume in each match, A has 0.55 chance to beat B, and 0.45 chance

to beat C, B has 0.55 chance to beat C.)

6) Let U be a geometric random variable with p = 0.3, V be a geometric

random variable with p = 0.3, X be a geometric random variable with p

= 0.5, Y be a geometric random variable with p = 0.4. Let U, V, X, Y

be independent and W = U + V + X + Y. Find P(W = k) for k = 5,6,7.

7) Let X,Y,Z be i.i.d. uniform random variables over the interval

[0,20]. Let U = min (X,Y,Z), V = max(X,Y,Z). Find E(U), E(V), Var(U),

Var(V), Cov(U,V), and correlation coefficient of U and V.

8) Do 100 experiments of problem 7, compute the sample mean, sample

variance of U and V. Also compute the sample covariance and sample

correlation coefficient of U and V. You have to state how you do your

experiments clearly and provide the raw data. Only take one decimal

point for your data.

9) The "fill" problem is important in industries. If an industry claims

that it is selling 16 ounces of its cereal in a box. If FDA finds out

more than 5% of its random sample (usually they randomly inspect 500

or more boxes) with weight < 16 ounces, then FDA will fine the company.

Suppose that the company's filling machine fills the cereal into the

cereal box according to N(a,O.9). What is the smallest a should be

so the company won't be fined. If it is N(16.05,b), what is the

smallest b should be so the company won't be fined.

12) The respective high school and college GPAS for 50 college seniors

as ordered pairs (x,y) are

(3.75, 3.19), (3.45, 3.34), (3.91, 3 65), (2.87, 2.43), (3.60, 3.46)

(3.42, 2.97), (2.94, 2.72), (4.00, 3.79), (2.65, 2.55), (3.10, 2.65)

(3.47, 3.15), (3.65, 3.39), (4.00, 3.86), (2.30, 2.12), (2.60, 2.43)

(2.47, 2.11), (3.36, 3.01), (3.60, 3.03), (3.65, 3.19), (3.46, 3.09)

(3.30, 3.05), (2.58, 2.63), (3.80, 3.42), (3.69, 3.27), (3.79, 3.47)

(3.27, 3.15), (3.55, 3.39), (4.00, 3.80), (2.40, 2.15), (2.80, 2.63)

(3.55, 3.39), (3.25, 3.44), (3.95, 3.75), (2.89, 2.63), (3.65, 3.56)

(3.47, 3.97), (2.94, 2.82), (4.10, 3.89), (2.65, 2.75), (3.15, 3.39)

(2.65, 2.79), (3.41, 3.25), (3.45, 3,59), (4.10, 3.96), (2.35, 2.62)

(2.60, 2.43), (2.47, 2.11), (3.36, 3.01), (3.60, 3.03), (3.65, 3.19)

(3.46, 3.09), (3.45, 3.29), (4.05, 3.89), (2.50, 2.45), (2.90, 2.73)

Find the best—fitting line and the sample correlation coefficient.

13) 400 faculty members at a state university are grouped according to

rank and sex as follows:

Instructor Assistant Professor Associate Professor Professor

Male 39 49 55 56

Female 48 60 48 42

If the significant level is 0.05, will you accept that the rank and

sex are indepedent? If your answer is not, then at what significant

level will you accept it? (This kind of problems is quite often

involved in the sex discrimination law cases.)

14) 50 students at the School participated in a wellness program to

lose weight for 6 weeks. The following are their weights before and

after the program:

(165, 157), (172, 167), (145, 142), (163, 164), (175, 168)

(156, 150), (148, 141), (139, 135), (134, 131), (168, 161)

(170, 165), (180, 172), (148, 140), (156, 157), (167, 160)

(145, 144), (150, 145), (170, 162), (156, 149), (149, 143)

(145, 139), (150, 147), (172, 165), (179, 173), (157, 155)

(156, 149), (163, 156), (175, 177), (176, 171), (166, 157)

(157, 148), (170, 162), (164, 157), (170, 172), (159, 157)

(163, 160), (174, 172), (168, 163), (176, 173), (157, 153)

(169, 173), (177, 172), (147, 141), (152, 159), (165, 160)

(145, 144), (150, 145), (170, 162), (156, 149), (149, 143)

We define the program is helpful if a participant loses at least

5.2 pounds after the program. If the significant level is 0.05, will

you accept that the program is helpful for losing weight? If not,

then at what significant level, you will accept that the program

is helpful? Compute the p—value.

15) Suppose we obtain the following BMI (Body Mass Index) of 40 males and 40 females.

Male: 25.4,23.5, 27.5,24.7, 26.8, 29.4, 27.8, 25.7, 23.5, 27.3,

23.6, 30.1, 26.7, 27.4, 24.6, 26.9, 27.3, 24.9, 26.8, 28.3,

26.9, 30.2, 25.9, 27.9, 26.5, 29.6, 25.3, 25.9, 28.6, 26.3,

28.4, 27.2, 25.3, 24.5, 27.3, 24.6, 30.2, 25.7, 27.2, 25.2

Female: 23.5, 24.3, 22.4, 23.6, 25.3, 26.3, 24.7, 25.9, 27.3, 26.3,

22.9, 27.3, 25.6, 26.3, 24.6, 25.3, 23.4, 23.7, 24.9, 22.9,

21.9, 24.5, 26.5, 25.3, 24.7, 25.8, 24.3, 23.8, 25.4, 22.6,

23.2, 24.1, 22.2, 22.6, 24.3, 23.7, 22.4, 23.2, 24.1, 22.3

At what significant level will you accept that in average a male’s

BMI is about 1.9 or more than a female's BMI? Compute the p—value.

17) Two players A and B agree to play the following coin tossing game:

A chooses string "HTTH" and B chooses string "HHTT". A fair coin is

tossed until either ”HTTH" or "HHTT” appear consecutively. If "HTTH"

appears before "HHTT" does, then A wins the game. If "HHTT" appears

before "HTTH" does, then B wins the game. What is probability that

A wins the game?

18) Let us classify the students into four classes: A&S (college

of Arts and Sciences), B (college of Business), E (college of

engineering), and O (others). Randomly ask 160 male students and 240

female students (under—graduates) and classify into these four classes.

If the significant level is 5%, will you accept the statement that the

male students and female students have the same distribution among

these four classes? If not, then at what significant level you will

accept the statement? There are 8 cells, each cell should have at least

12 observations. (randomly generate!)

19) A pharmaceutical salesman claimed that the side effect of their new drug

is at most 5%. If you give the drug to 10,000 patients and 546 patients

are getting side effect. If the significant level is 5%, will you accept

her (or his) claim? lf not, then at what significant level you will accept

her (or his) claim? Also compute the p—value.

21)Let S, T, U be i.i.d. exponential distribution random variables with

Θ = 10. Let X = min(S,T,U) and Y = max(S,T,U). Find E(X), E(Y),

Var(X), Var(Y) and Cov(X,Y).

22) Let X, Y, and Z be three random variables distributed uniformly over

the interval [O,1O]. Let A be the event that X, Y, and Z form a

triangle, ie., X + Y > Z, X + Z > Y, and Y + Z > X. Prove that P(A)

= 0.5.

23)Let X and Y be two random variables distributed uniformly over

the interval [O,1O]. They split the interval [O,1O] into three

parts, a, b, C. Let B be the event that a, b, and c form a triangle,

ie., a + b > x, a + c > b, and b + c > a. Prove that P(B) : 0.25;

Subject | Mathematics |

Due By (Pacific Time) | 12/13/2013 12:00 am |

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