Midterm Flashcards
What is a probability model?
sample space Ω
collection of event F
probability measure P
What can you say if A1, A2, … , An are pairwise disjoint?
P(A1 U A2 U … U A3) = P(A1) + P(A2) + … + P(An)
How can you find P(A)
P(A) = #A / #Ω
What is F?
all possible subsets of Ω
what is P({w})
1 / #Ω
What is the binomial coefficient?
number of possibilities to choose k objects from n if the order is irrelevant
(n k) = n! / k!(n-k)!
What is a discrete sample space?
Sample space with a finite or countably infinite number of outcomes
What is a continuous sample space?
A sample space with uncountably many outcomes
What are the properties of complements?
A U Ac = Ω
P(A) + P(Ac) = 1
P(A) = 1 - P(Ac)
What does monoticity says?
A included in B –> P(A) <= P(B)
P(B) = P(A) + P(B∩Ac)
else
P(B) = P(A∩B) + P(Ac∩B)
What is the inclusion exclusion formula?
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)
What is a random variable?
A random variable is a function from Ω–>R
What is a probability mass function?
The probability distribution for discrete random variables
P(k) = P(X=k)
When can X be called degenerate?
if there exists b with P(X∈b)=1
What is conditional probability? What if Ω is finite and every outcome has the same probability?
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
What is the multiplication rule?
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)
What is the law of total probability?
{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)What is Bayes’ Formula
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)
When are two events independent? What does it imply?
When P(A∩B) = P(A)*P(B)
also,
Ac and B, A and Bc, Ac and Bc are independent
When are three sets A, B and C independent?
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)
What is P(A | B) equal to if A and B are independent?
P(A | B) = P(A) * P(B) / P(B) = P(A)
P(B | A) = P(B)
When are A1, A2, … , An conditionally independent given B?
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)
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?
B1,…,Bk are also independent
What is a probability density function?
The probability distribution of a continuous random variable
P(X<=b) = ∫(-oo to b) f(x) dx
How do you find the density function from the cumulative distribution function?
If F is continuous and F’ exists then
f(x) = F’(x)
How do you find P(a
P(a
What is the Expectation?
discrete: E(g(x)) = Σ(k) g(k)P(X=k)
continuous: E(g(x)) = ∫(-oo to oo) g(x)f(x) dx
What is the nth moment of the random variable x?
expectation E(x^n)
What is the median of a random variable x?
any real value m st P(X>=m) >= 1/2 and P(X<=m) >= 1/2
what is the pth quantile of a random variable x?
any real value x st P(X>=x) >= 1-p and P(X<=x) >= p
What are the variance and the standard deviation of a random variable x?
Var(X) = E(X^2) - (E(X))^2 stdev = sqrt(Var(X))
if Var(X) = 0 then P(X=a) = 1
