Probability Flashcards
The set of all possible outcomes is what?
The sample space Ω
Possible results of an experiment are called what?
Outcomes
What is an event?
A subset of the sample space
What does probability express?
How likely an event is
Three basic properties of probabilities
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)
What is the complement of A in sample set theory?
Not A
What is the intersection of A and B in sample set theory?
Both A and B
What is the union of A and B in sample set theory?
Either A or B
What does it mean for two events to be mutually exclusive?
A intersect B = 0
What does P(A’) equal?
P(A’) = 1 - P(A)
What does P(A union B) equal?
P(A union B) = P(A) + P(B) - P(A intersect B)
How would the conditional probability of A given that B be denoted?
P(A|B)
What does P(A|B) equal?
P(A intersect B)/P(B)
Multiplication rule
P(A intersect B) = P(A|B)P(B) = P(B|A)P(A)
When are two events A and B independent?
P(A intersect B) = P(A)P(B)
Bayes’ theorem
P(A|B) = P(B|A)P(A)/P(B)
What is the difference between permutations and combinations?
Permutations are order-specific
Combinations are groups where the ordering doesn’t matter
What does nPr equal?
nPr = n!/(n-r)!
What does nCr equal?
n!/((n-r)!r!)
What is the mean of a discrete random variable?
⟨X⟩ = Sum of (x(j) P(X=x(j))) between N and j=1
What is the variance of a discrete random variable?
var(X) = ⟨(X - ⟨X⟩)^2⟩ = Sum of ((x(j) - ⟨X⟩)^2 P(X=x(j)))
For the binomial distribution B(n, p), what is P(X=r) equal to?
P(X=r) = nCr p^r (1-p)^(n-r)
What is the mean of B(n, p) equal to?
mean of B(n, p) = np
What is the variance of B(n, p) equal to?
var(B(n, p)) = np(1-p)
What is P(X=r) equal to in Poisson distribution?
P(X=r) = e^(-λ) (λ^r)/r!
What are the mean and variance of the Poisson distribution equal to?
λ = var(X) = ⟨X⟩
What is the main feature of a continuous variable X?
X has a probability density function f(x) defined such that the probability of finding X between x - dx/2 and x + dx/2
For a continuous variable, X, what is P(a =< X =< b)?
Integrand of f(x) dx between b and a
Define the cumulative distribution function F(a)?
F(a) = P(X=<a></a>
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
⟨X⟩ = integrand of x f(x) dx between +/- infinity var(X) = integrand of (x-⟨X⟩)^2 f(x) dx between +/- infinity
Considering a continuous random variable X uniformly distributed between 0 and 1, what are the values of the mean and variance?
⟨X⟩ = 1/2 var(X) = 1/12
What is f(x) for Normal distribution?
f(x) = 1/sqrt(2πσ^2) e^(-((x-μ)^2)/(2σ^2))
What is the mean value and the variance value for Normal distribution?
⟨X⟩ = μ var(X) = σ^2
What is cumulative distribution of Normal distribution equal to?
F(a) = 1/2 [1 + erf((a-μ)/sqrt(2σ^2))
