1. Give a recursive definition for the sequence: 2, 4, 8, 16, 32 ?

2. Let f(x) = -2f(x-1) + 5 and f(0) = 2, find f(4).

3. In New York City, there are two non-bald people who have the same number of hairs on their head. (The human head can contain up to several hundred thousand hairs, with a maximum of about 500,000.)

4. Imagine BMCC has 25,000 students, at least one from each of the 50 states. Then there must be a group of 500 students coming from same state.

5 & 6 : For each situation, decide if the random variable described is a discrete random variable or a continuous random variable.

5. Random variable X = the number of letters in the last name of a student picked at random from a class on English composition.

1. Discrete random variable
2. Continuous random variable

6. Random variable X = the time (seconds) it takes one email to travel between a sender and receiver.

1. Discrete random variable
2. Continuous random variable

7 to 9 : The probability distribution for X = number of heads in 4 tosses of a fair coin is given in the table below.

7. What is the probability of getting at least one head?

1. 1/16
2. 4/16
3. 5/16
4. 15/16

8. What is the probability of getting 1 or 2 heads?

1. 4/16
2. 6/16
3. 10/16 D.14/16

9. What is the value of the cumulative distribution function at 3, i.e. P(X ? 3)?

1. 6/16
2. 10/16
3. 11/16 D.15/16

10. For the relation on the set {1, 2, 3, 4}, decide whether it is reflexive, whether it is symmetric, whether it is anti-symmetric, and whether it is transitive. In order to get full credits: For example, if it?s not reflexive, you must state why it is not. {(1, 1), (1, 2), (2, 3), (2, 4), (3, 2), (3, 3)}

Reflexive:

Symmetric:

Anti-symmetric:

Transitive:

K P(X= k) 0 1 1/16 4/16 2 6/16 3 4/16 4 1/16