Midterm

Question 1

What is a probability model?

Answer 1

sample space Ω
collection of event F
probability measure P

Question 2

What can you say if A1, A2, … , An are pairwise disjoint?

Answer 2

P(A1 U A2 U … U A3) = P(A1) + P(A2) + … + P(An)

Question 3

How can you find P(A)

Answer 3

P(A) = #A / #Ω

Question 4

What is F?

Answer 4

all possible subsets of Ω

Question 5

what is P({w})

Answer 5

1 / #Ω

Question 6

What is the binomial coefficient?

Answer 6

number of possibilities to choose k objects from n if the order is irrelevant
(n k) = n! / k!(n-k)!

Question 7

What is a discrete sample space?

Answer 7

Sample space with a finite or countably infinite number of outcomes

Question 8

What is a continuous sample space?

Answer 8

A sample space with uncountably many outcomes

Question 9

What are the properties of complements?

Answer 9

A U Ac = Ω
P(A) + P(Ac) = 1
P(A) = 1 - P(Ac)

Question 10

What does monoticity says?

Answer 10

A included in B –> P(A) <= P(B)
P(B) = P(A) + P(B∩Ac)

else
P(B) = P(A∩B) + P(Ac∩B)

Question 11

What is the inclusion exclusion formula?

Answer 11

P(AUB) = P(A) + P(B) - P(A∩B)
for 3 events:
P(AUBUC) = P(A) + P(B) +P(C) - P(A∩B) - P(A∩C) - P(B∩C) + P(A∩B∩C)

Question 12

What is a random variable?

Answer 12

A random variable is a function from Ω–>R

Question 13

What is a probability mass function?

Answer 13

The probability distribution for discrete random variables

P(k) = P(X=k)

Question 14

When can X be called degenerate?

Answer 14

if there exists b with P(X∈b)=1

Question 15

What is conditional probability? What if Ω is finite and every outcome has the same probability?

Answer 15

P(A | B) = P(A∩B) / P(B)
P(A1 U A2 | B) = P(A1 | B) + P(A2 | B) - P(A1∩A2 | B)

then
P(A | B)= P(A∩B) / P(B) = #A∩B / #B

Question 16

What is the multiplication rule?

Answer 16

2 events:
P(A∩B) = P(A)P(B | A)
3 events:
P(A∩B∩C)= P(A)P(B | A)*P(C | A∩B)

Question 17

What is the law of total probability?

Answer 17

{B1,...,Bn} partition of Ω,
if Bi pairwise disjoint 
and U(from i=1 to n) of Bi = Ω
then
P(A) = Σ (from i=1 to n) of P(A∩Bi) =  Σ (from i=1 to n) of P(Bi)*P(A | Bi)

Question 18

What is Bayes’ Formula

Answer 18

If P(A), P(B), P(Bc) >0, then
P(B | A) = P(A∩B) / P(A) = P(B)*P(A | B) / P(A∩B)+P(A∩Bc)

and P(A∩B) = P(B)*P(A | B)
and P(A∩Bc) = P(Bc)*P(A | Bc)

Question 19

When are two events independent? What does it imply?

Answer 19

When P(A∩B) = P(A)*P(B)
also,
Ac and B, A and Bc, Ac and Bc are independent

Question 20

When are three sets A, B and C independent?

Answer 20

If
P(A∩B) = P(A) * P(B)
P(A∩C) = P(A) * P(C)
P(B∩C) = P(B) * P(C)
P(A∩B∩C) = P(A) * P(B) * P(C)

Question 21

What is P(A | B) equal to if A and B are independent?

Answer 21

P(A | B) = P(A) * P(B) / P(B) = P(A)

P(B | A) = P(B)

Question 22

When are A1, A2, … , An conditionally independent given B?

Answer 22

if for any k∈{2,…,n} and 1<=i1<=i2<=…<=ik<=n

P(Ai1∩Ai2∩…∩Aik | B) = P(Ai1 | B) * P(Ai2 | B) * … * P(Aik | B)

Question 23

If A1,…,An independent, what can you say about B1,…,Bk, which are constructed from Ais st two different Bjs do not use the same Ais?

Answer 23

B1,…,Bk are also independent

Question 24

What is a probability density function?

Answer 24

The probability distribution of a continuous random variable

P(X<=b) = ∫(-oo to b) f(x) dx

Question 25

How do you find the density function from the cumulative distribution function?

Answer 25

If F is continuous and F’ exists then

f(x) = F’(x)

Question 26

How do you find P(a

Answer 26

P(a

Question 27

What is the Expectation?

Answer 27

discrete: E(g(x)) = Σ(k) g(k)P(X=k)
continuous: E(g(x)) = ∫(-oo to oo) g(x)f(x) dx

Question 28

What is the nth moment of the random variable x?

Answer 28

expectation E(x^n)

Question 29

What is the median of a random variable x?

Answer 29

any real value m st P(X>=m) >= 1/2 and P(X<=m) >= 1/2

Question 30

what is the pth quantile of a random variable x?

Answer 30

any real value x st P(X>=x) >= 1-p and P(X<=x) >= p

Question 31

What are the variance and the standard deviation of a random variable x?

Answer 31

Var(X) = E(X^2) - (E(X))^2
stdev = sqrt(Var(X))

if Var(X) = 0 then P(X=a) = 1