Prob A

Question 1

What is a sample space?

Answer 1

The set whose elements each correspond to a possible outcome of the experiment, so that every possible outcome is included

Question 2

What is an event?

Answer 2

A property that the outcome of the experiment may or may not have. We can identity the elements of the sample space with the relevant property

Question 3

Give the notation for

i) A and B
ii) A or B
iii) not A
iv) A is a subset of B

Answer 3

Question 4

What is a probability measure

Answer 4

Question 5

What does it mean for A and B to be mutually disjoint?

Answer 5

The intersection of A and B is empty

Question 6

Prove that P(A) + P(A^c) = 1

Answer 6

Question 7

Prove the relationship between P(A) and P(B) if A is a subset of B

Answer 7

Question 8

Prove that P(AuB) = P(A) + P(B) - P(AnB)

Answer 8

Question 9

State the Inclusion-Exclusion formula for events not necessarily disjoint

Answer 9

Question 10

What is the total number of permutations of {1,2,….,n}

Answer 10

n!

Question 11

Lemma; Let S be a set having n elements and let 0<=k<=n. The number of subsets of S having exactly k elements is nCk = n!/(k! x (n-k!))

Answer 11

Question 12

State the Fundamental Multiplication rule

Answer 12

Suppose a procedure has k steps with n₁ ways of performing the 1st step, n₂ ways of performing the second step, up to n_k ways of performing the kth step. Then the total number of different ways is n₁ x n₂ x …. x n_k

Question 13

Suppose a population of size N=K₁ + K₂ contains K₁ individuals of type 1 and K₂ individuals of type 2. If sample of size n is drawn with replacement from this population it contains k₁ individuals of type 1 and k₂ individuals of type 2. Whats the probability

Answer 13

Question 14

Suppose a population of size N=K1 + K2 contains K1 individuals of type 1 and K2 individuals of type 2. If sample of size n is drawn without replacement from this population it contains k1 individuals of type 1 and k2 individuals of type 2. Whats the probability

Answer 14

Question 15

Give the conditional probability of P(B | A) and state the important conditions

Answer 15

Question 16

State and Prove the multiplication rule for conditional probabilities

Answer 16

Question 17

When are events a partition of a subspace

Answer 17

Question 18

State the law of total probability

Answer 18

Question 19

Prove the total law of probability

Answer 19

Question 20

State Bayes Formula

Answer 20

Question 21

Prove Bayes formula

Answer 21

Question 22

When are events E_1,E₂,…..,E_n mutually independent

Answer 22

When for all choices of 1 to n

P(E₁ n E₂ n …. n E_n) = P(E₁) x P(E₂) x … x P(E_n)

Question 23

When does a random variable X have the binomial distribution with parameters n and p

Answer 23

Question 24

Prove the binomial probability distribution

Answer 24

Question 25

State the law of large numbers for the Binomial distribution

Answer 25

Question 26

State the binomial to poisson distribution theorem

Answer 26

Question 27

Prove the conversion of binomial to poission

Answer 27

Question 28

What is the most likely outcome of a experiment with n repeats and probability of success p

Answer 28

np

Question 29

State the Theorem for the gaussian approximation to the normal distribution

Answer 29