Probability

Question 1

The set of all possible outcomes is what?

Answer 1

The sample space Ω

Question 2

Possible results of an experiment are called what?

Answer 2

Outcomes

Question 3

What is an event?

Answer 3

A subset of the sample space

Question 4

What does probability express?

Answer 4

How likely an event is

Question 5

Three basic properties of probabilities

Answer 5

0 =< P(A) =< 1 for all events A
P(Ω) = 1
If events A and B have no outcomes in common, the probability of either is P(A) + P(B)

Question 6

What is the complement of A in sample set theory?

Answer 6

Not A

Question 7

What is the intersection of A and B in sample set theory?

Answer 7

Both A and B

Question 8

What is the union of A and B in sample set theory?

Answer 8

Either A or B

Question 9

What does it mean for two events to be mutually exclusive?

Answer 9

A intersect B = 0

Question 10

What does P(A’) equal?

Answer 10

P(A’) = 1 - P(A)

Question 11

What does P(A union B) equal?

Answer 11

P(A union B) = P(A) + P(B) - P(A intersect B)

Question 12

How would the conditional probability of A given that B be denoted?

Answer 12

P(A|B)

Question 13

What does P(A|B) equal?

Answer 13

P(A intersect B)/P(B)

Question 14

Multiplication rule

Answer 14

P(A intersect B) = P(A|B)P(B) = P(B|A)P(A)

Question 15

When are two events A and B independent?

Answer 15

P(A intersect B) = P(A)P(B)

Question 16

Bayes’ theorem

Answer 16

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

Question 17

What is the difference between permutations and combinations?

Answer 17

Permutations are order-specific

Combinations are groups where the ordering doesn’t matter

Question 18

What does nPr equal?

Answer 18

nPr = n!/(n-r)!

Question 19

What does nCr equal?

Answer 19

n!/((n-r)!r!)

Question 20

What is the mean of a discrete random variable?

Answer 20

⟨X⟩ = Sum of (x(j) P(X=x(j))) between N and j=1

Question 21

What is the variance of a discrete random variable?

Answer 21

var(X) = ⟨(X - ⟨X⟩)^2⟩ = Sum of ((x(j) - ⟨X⟩)^2 P(X=x(j)))

Question 22

For the binomial distribution B(n, p), what is P(X=r) equal to?

Answer 22

P(X=r) = nCr p^r (1-p)^(n-r)

Question 23

What is the mean of B(n, p) equal to?

Answer 23

mean of B(n, p) = np

Question 24

What is the variance of B(n, p) equal to?

Answer 24

var(B(n, p)) = np(1-p)

Question 25

What is P(X=r) equal to in Poisson distribution?

Answer 25

P(X=r) = e^(-λ) (λ^r)/r!

Question 26

What are the mean and variance of the Poisson distribution equal to?

Answer 26

λ = var(X) = ⟨X⟩

Question 27

What is the main feature of a continuous variable X?

Answer 27

X has a probability density function f(x) defined such that the probability of finding X between x - dx/2 and x + dx/2

Question 28

For a continuous variable, X, what is P(a =< X =< b)?

Answer 28

Integrand of f(x) dx between b and a

Question 29

Define the cumulative distribution function F(a)?

Answer 29

F(a) = P(X=

Question 30

Mean and variance for a discrete variable but P(X=x(j))->f(x) dx and Sum between N and j=1 ->integrand between +/- infinity

Answer 30

``` ⟨X⟩ = integrand of x f(x) dx between +/- infinity var(X) = integrand of (x-⟨X⟩)^2 f(x) dx between +/- infinity ```

Question 31

Considering a continuous random variable X uniformly distributed between 0 and 1, what are the values of the mean and variance?

Answer 31

``` ⟨X⟩ = 1/2 var(X) = 1/12 ```

Question 32

What is f(x) for Normal distribution?

Answer 32

f(x) = 1/sqrt(2πσ^2) e^(-((x-μ)^2)/(2σ^2))

Question 33

What is the mean value and the variance value for Normal distribution?

Answer 33

``` ⟨X⟩ = μ var(X) = σ^2 ```

Question 34

What is cumulative distribution of Normal distribution equal to?

Answer 34

F(a) = 1/2 [1 + erf((a-μ)/sqrt(2σ^2))