Probability A //

Question 1

Experiment

Answer 1

Any process in which the outcome is uncertain and cannot be predicted in advance

Question 2

Sample space (Ω)

Answer 2

A set where elements - called sample points - each correspond to a possible outcome of the experiment

Question 3

Event

Answer 3

A property the outcome of the experiment may or may not have

Question 4

Random variable

Answer 4

Any quantity we may measure whose observed value depends on the outcome of the experiment

Question 5

Probability P(A)

Answer 5

(number of sample points in A)/(number of sample points in Ω)

Question 6

Probability measure

Answer 6

A function A → P(A) for events A ⊆ Ω satisfying:
1. for every event for which P(A) is defined, 0 ≤ P(A) ≤ 1, with P(∅) = 0 and P(Ω) = 1
2. for any finite or countably infinite sequence or mutually disjoint events A1, A2, A3, … with every P(Ai) defined we have
P(∪(i=1,n)Ai) = Σ(i=1,n)P(Ai)
P(∪(i=1,∞)Ai) = Σ(i=1,∞)P(Ai)
(additive property)

Question 7

Binomial distribution B(n,p)

Answer 7

If X has this distribution, then P(X=k) = (n k)p^k(1-p)^(n-k) for k=0,1,…,n

Question 8

Poisson distribution Po(λ)

Answer 8

If X has this distribution, then P(X=k) = e^(-λ)λ^k/k!

Question 9

Normal density function

Answer 9

φ(x) = e^(-x^2/2)/√(2π)

Question 10

Stirling’s n! formula

Answer 10

n! ~ √(2πn)n^ne^-n

Question 11

Conditional probability of B given A, P(A) ≠ 0

Answer 11

P(B|A) = P(A ∩ B)/P(A)

Question 12

Law of total probability

Answer 12

If A1,…,An is a partition of a sample space Ω with P(Ai) > 0 for each I, then for any event B ⊂ Ω we have P(B) = Σ(i=1,n) P(Ai)P(B|Ai)

Question 13

Bayes formula

Answer 13

If A1,…,An is a partition of Ω with P(Ai)>0 for all i, and B a further event with P(B)>0, then for each j
P(Aj|B) = P(B|Aj)*P(Aj)/(Σ(i=1,n)P(B|Ai)P(Ai))

Question 14

Independent

Answer 14

Two events A and B are called independent if knowledge A has occurred does not affect the probability of B occurring (and vice versa)

Question 15

Mutually independent events

Answer 15

Events E1,E2,…,En,… are mutually independent if for all choices of i1,i2,..,in all distinct, then P(Ei1 ∩ Ei2 ∩ … ∩ Ein) = P(Ei1) * P(Ei2) * … P(Ein)

Question 16

Mutually independent rvs

Answer 16

R.vs X1, X2, …, Xn, … are mutually independent r.vs if whenever B1, B2, …, Bn, … are subsets of ℝ the events {X1 ∈ B1}, {X2 ∈ B2), …, {Xn ∈ Bn} are mutually independent events

Question 17

Partition of a sample space

Answer 17

(Ai) is a partition if ∪(i=1,n)Ai = Ω, Ai ∩ Aj = ∅ if i ≠ j

Question 18

Mutually disjoint events A1, A2,…

Answer 18

Ai ∩ Aj = ∅ for i ≠ j

Question 19

Fundamental multiplication rule for counting elements in a set

Answer 19

A number k choices are to be made. if there are m1 possibilities for the first choice, m2 possibilities for the second choice through to mk for the kth choice, and those may be combined freely, then the total number of possibilities for the procedure is m1m2…*mk

Question 20

Weak law of large numbers

Answer 20

Fix p ∈ (0,1), let X have the binomial dist with parameters n and p, then for any ε > 0, P(np - nε < X < np + nε) → 1 as n → ∞

Question 21

Properties of any prob measure

Answer 21

1. P(Aᶜ) = 1 - P(A)
2. If A ⊂ B then P(A) ≤ P(B)
3. For all A,B,
   P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
4. inclusion-exclusion formula for any Ai:
   P(∪(i=1, n)Ai) = Σ(i=1, n)P(Ai) -
   Σ(i1

Question 22

Central limit theorem for binomial

Answer 22

Fix p ∈ (0,1), let Xn have the binomial dist with parameters n and p. Then as n → ∞
P(np + z1√(np(1-p)) ≤ Xn ≤ np + z1√(np(1-p)))
→ ∫(z1, z2)φ(x)dx