Take-home Problems for Exam 1

Open book, open notes. You may work together with others. Justify all answers with neat and organized work on your own paper. Clearly indicate your answers.

5 points each problem, 20 points possible. The other 80 points will be on the in-class exam, Thursday, October 2.

Due: Friday, October 3, at 1:00 p.m. Stiff penalty for late submissions. Start now!

1. How many ways are there to distribute nine different books among 15 different children if no child gets more than one book?

2. How many arrangements of JUPITER are there with the vowels in alphabetic order?

3. How many numbers greater than 3,000,000 can be formed by arrangements of 1, 2, 2, 4, 6, 6, 6?

4. How many arrangements of MISSISSIPPI are there with no consecutive S's?

Math 421

Exam 1

Justify all answers with neat and organized work. Clearly indicate your answers. 100 points possible.

1. (20 pts.) Don't forget the Take Home Problems, due tomorrow.

2. (10 pts.) Let X equal the number of spots on the side that is up after one ordinary six-sided die is cast at random. Thus, the space of X is S={1,2,3,4,5,6}.

(a) If the experiment is fair and unbiased, assign reasonable probabilities to these 6 outcomes. Then compute the mean m of this probability distribution.

(b) This experiment was repeated 20 times, and here are the results.

3    6    5    4    3    5    4    1    1    6    5    2    1    5    5    2    5    6    4    5

Using these data, (i) construct a frequency table, (ii) draw a relative frequency histogram, and (iii) find the sample mean [`x].

3. (10 pts.) Consider a random variable X which has p.d.f.

f(w)=4x3,       0 £ w < 1.

Use the formulas

m = ó õ S  xf(x) dx

and

s2= ó õ S  x2 f(x) dx -m2

to find the mean and variance of this distribution. Also find p_0.2, the 20th percentile.

4. (10 pts.) Let P be a probability measure such that

P(A)= 1 3 , P(B)= 1 2 , and P(AÈB)= 2 3 .

Find each of these.

(a) P(AÇB)

(b) P(AÇB¢)

(c) P(A¢ÈB¢)

(d) P(A¢ÈB)

5. (5 pts.) Determine the conditional probability that a sum of 7 is obtained on the toss of a pair of ordinary fair dice, given that the sum is odd.

6. (5 pts.) The ``face cards'' an ordinary 52-card deck are the jacks, queens, and kings. Suppose two different cards are selected at random. (``Different'' means ``without replacement.'') What is the conditional probability that a pair of kings was obtained, given that the 2 cards were face cards?

7. (5 pts.) Give the mathematical definition of what it means for events A and B to be independent.

8. (5 pts.) Prove that if A and B are independent, then so are A and B¢.

9. (10 pts.) Bowl I contains 3 red chips and 7 blue chips. Bowl II contains 6 red chips and 4 blue chips. A bowl is selected at random and then 1 chip is drawn from this bowl.

(a) Compute the probability that this chip is red.

(b) Given that the chip is red, compute the conditional probability that this chip was drawn from bowl II.

10. (10 pts.) For each of the following, determine the constant c so that f(x) satisfies the conditions of being a p.m.f. for a random variable X.

(a)

f(x)=x/c,       x=1,2,3,4,5.

(b)

f(x)=c( 2 3 )x,       x=1,2,3,....

11. (10 pts.) The random variable X has p.m.f.

f(x)= ì ï ï ï ï í ï ï ï ï î 1 6 ,     x=0, 2 2 3 ,     x=-2.

Find each of these.

(a) E[2(X²+1)²]

(b) E[2(X+1)²]

(c) E[cos[(pX)/4]]