1   Multiple Choice Questions

Pleasefill in the bubble(s) on the exam below corresponding to your answer. You do not need to submit any additional workfor these questions.

1. Given a probability space (Ω, F, P) and three events A, B, C ∈ F, which of the following correctly computes P(A1 ∪ A2 ∪ A3)?

⃝  P(A1) + P(A2) + P(A3)

^  1 − P(A1(C)) · P(A2(C)  | A 1(C)) · P(A3(C)  | A 1(C) ∩ A2(C))

⃝  P(A1) · P(A2) · P(A3)

⃝  1 − P(A1) − P(A2) − P(A3)

⃝  None of the other answer choices

2. Consider a random variable X with p.m.f. given by k   −2     1

Which of the following is the correct value of E[X]?

^  0

⃝  1/3

⃝  2/3

⃝  1

⃝  2

⃝  None of the above

3. Fill in the Blanks: Discrete random variables have state spaces that are                  ,     [1pts.]

whereas continuous random variables have state spaces that are                   .

⃝  finite; infinite

⃝  countable; uncountable

^  at most countable; uncountable

⃝  uncountable; at most countable

⃝  uncountable; countable

⃝  None of the above.

4. In a bag of 100 marbles, 40 are blue and the remaining 60 are gold. Yaz draws marbles one by one at random, replacing the marble each time. If X denotes the number of marbles (including the final marble) Yaz has to draw before she observes her 3rd blue marble, which of the following accurately describes the distribution of X?

⃝  Bern(40)

⃝  Bern(0.4)

⃝  Bin(3, 0.4)

⃝  NegBin(40, 0.4)

^  NegBin(3, 0.4)

⃝  HyperGeom(40, 100, 3)

⃝  Poisson(0.4)

⃝  None of the above.

5. Which of the following statements is true in general?                 ⃝  Pairwise independence implies mutual independence.

^  Two mutually dependent events can be conditionally independent

⃝  There are 2n computations needed to establish the mutual indepen- dence of n events

⃝  Pairwise independence is a stronger condition than mutual indepen- dence.

⃝  All of the above answer choices are false.

2   Short Answer Questions

Please mark yourfinal answers in the spaces provided below each question. Be sure to show all of your work!

6. Consider a probability space (Ω, F, P) and suppose A and B are two events. Prove the identity

P(A / B) = P(A) · P(BC  | A)

7. The management of GauchoStay apartments is quite lazy, and has allowed an ant infestation to manifest. There is a 10% chance that a randomly selected unit will have an infestation problem, independently of all other units. The exter- minator goes from unit to unit, but is forgetful and could visit the same unit twice. For this problem, there is no need to simplify your answers.

(a) What is the probability that the exterminator observes exactly 3 infested

units among the first 7 units they examine?

(b)

What is the probability that the 12th unit the exterminator examines is the third infested unit they observed?

(c)

What is the expected number of units the exterminator must visit before observing their second infested unit?

[2pts.]

(d)

Now, suppose that there are 100 units in GauchoStay and 10 of them are infested.  Additionally, suppose that the exterminator now takes care to not examine the same apartment twice.  What is the probability that the exterminator observes exactly 3 infested units in a sample of 6 units?

[2pts.]

8. Let X be a continuous random variable with probability density function (p.d.f.)

given by

fX (x) = { · x

if 0 ≤ x ≤ 3

otherwise

(a)

Verify that fX (x) is a valid probability density function.

(b) Compute

E [ ]

Show all of your steps, including any integration you perform!

(c)

Find FX(x), the cumulative distribution function (c.d.f.) of X. Be sure to consider all cases!

[4pts.]

9. In a drawer, you have 2 red socks, 2 white socks, and 2 green socks. You ran- domly draw a sample of 4 socks, without replacement; let X denote the number of matching pairs in your sample (by matching, we mean in color).

(a) What is the state space SX of X?

Find the probability mass function of X.