IP

Question 0

What axioms does a probability function satisfy?

Answer 0

(1) P(A)>=0
(2) P(sample space)=1
(3) If A1,A2,… are mutually exclusive members of F (Ai N Aj= empty set) then P(U from i=1 to infinity of Ai)= E from i=1 to infinity of P(Ai).
Though if the sample space is finite, then (3) becomes P(AUB)=P(A)+P(B) if A N B = empty set

Question 1

A collection of events, F, is called a sigma-field if it satisfies which axioms?

Answer 1

(1) sample space is in F
(2) if A is in F then A complement is in F
(3) if A1,A2,… is in F then so is A1UA2U…

Question 2

How can these be written:

1) P(A^c
(2) P(AUB)
(3) P(empty set)

Answer 2

(1) P(A^c)=1-P(A)
(2) P(AUB)=P(A)+P(B)-P(A N B)
(3) P(empty set)=0

Question 3

What is the equation for conditional probability and what is Bayes’ rule?

Answer 3

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

Bayes’ rule: P(A|B)=P(B|A)P(A)/P(B)

Question 4

Two events are said to be independent if and only if…?

Answer 4

P(A N B)=P(A)P(B)
and then…
P(A|B)=P(A) and P(B|A)=P(B)

Question 5

The cumulative distribution function is defined by what?

Answer 5

F(x)=P(X<=x)

Question 6

What are the properties of a c.d.f?

Answer 6

1. F(X1)<=1
2. Lim F(x) as x goes to infinity is 1
3. Lim F(x) as x goes to -infinity is 0
4. Lim F(x) as x goes to a+ = F(a) (right continuous)

Question 7

What is a probability mass function defined by and what are its properties?

Answer 7

Pmf relates to discrete random variables and is defined as f(x)=P(X=x).

1. The set of x such that f(x) is not equal to 0 is countable.
2. Sum from k in real numbers of f(x)=1 for all x where f(x) is not zero.

Question 8

What is a probability density function and what are its properties?

Answer 8

A probability density function relates to continuous random variables and is defined as F(x)=integral from -infinity to x of f(x)dx=1

1. P(X=x)=0
2. Integral from -infinity to infinity of f(x)dx=1
3. P(a<=b)=integral from a to b of f(x)dx

Question 9

What is the expectation and variance for:

1. Discrete random variables
2. Continuous random variables

Answer 9

1. EX: sum from k of kP(X=k)
   VarX: EX^2 - (EX)^2 where EX^2=k^2P(X=k)
2. EX: integral of xf(x)dx
   VarX: EX^2 - (EX)^2 where EX^2=the integral of x^2f(x)dx

Question 10

What is a Binomial, Geometric and Poisson distribution used for?

Answer 10

Binomial: n independent Bernoulli trials are performed and X represents the number of successes that occur in n trials.
Geometric: probability p of success. The sequence is observed until the first success occurs. X represents the trial number of the first success.
Poisson: can be used with approximating binomial r.v. parameters (n,p) where n is large and p is small so that np is a moderate size.