Probability Flashcards
Define the population, a sample an event and a singleton.
population: set of all entities under study
sample: an element of the sample space: the set of all outcomes
event: subset of the sample space
singleton: an individual element of the sample space
Define the probability measure and give the three axioms
P : \Omega \to [0, 1]
P(Omega) = 1
P(A U B) = P(A) + P(B) if A n B = 0
For a finite sample space Ω, suppose that each singleton is equally likely. What is the probability of an event A?
A / # Omega
Compare the probability of a union of two events A and B to the individual probabilities for those events individually. What about intersections?
P(A U B) >= P(A) or P(B)
P(A n B) <= P(A) or P(B)
Give the formula for the probability of event A conditional on event B.
P(A|B) = P(A n B)/P(B)
State Bayes’ theorem. What is this useful for?
P(A|B) = P(B|A)P(A)/P(B)
This is useful for reversing conditionals
Define independence.
P(A|B) = P(A) i.e. P(A n B) = P(A)P(B)
What is a random variable?
A map X : \Omega -> IR subject to randomness
Give an example of a random variable and a variable that is not random.
Result of a dice throw.
The number of players on a football pitch
What is the probability a random variable X is in some subset S of the real numbers?
P(X \in S) = P({\omega \in Omega : X(\omega) \in S}
Give an example of a finite and an infinite discrete random variable.
Result of a dice throw.
random choice of integer
Define the CDF
F_X(z) = P(X <= z)
Define the PMF. What type of random variable is this used for?
f_X(z) = P(X = z)
This is for discrete random variables
Define the PDF. What type of random variable is this used for?
P(a <= z <= b) = \int_a^b dz f_X (z)
Give some examples of normally distributed random variables.
heights of females
test scores
Give some examples of Poisson distributed random variables.
goals in football matches
number of bones a person has broken
Define the joint PMF/PDF for two random variables X and Y.
P(X \in [a,b], y \in [c,d]) = \int_a^b \int_c^d dz1 dz2 f_(X, Y) (z1, z2)
How does this simplify if X and Y are independent?
f_(X,Y)(z1,z2) = f_X(z1)f_Y(z2)
What is the PMF/PDF for a random variable X conditional on another random variable Y?
P(X \in [a,b] | Y \in [c,d]) = \int a^b dz f_(X|Y \in [c,d])
Define the expectation of a discrete and a continuous random variable.
E[X] = \sum_x x f_X(x) E[X} = \int_-infty^infty x f_X (x)
What is the expectation of a function of a random variable?
E[g(X)] = \int_(-\infty)^in\fty g(x) f_X(x)
What is the variance of a random variable and what does it measure?
var(X) = E[(X-E[X])^2]
measures expected squared deviation from the expectation
What is the covariance between two random variables X and Y? What does it measure?
cov(X,Y) = E[(X-E[X])(Y-E[Y])]
measures the extent to which the two variables are linearly related
Why can the covariance sometimes be hard to interpret.
It is dependent on the scale of X and Y
Define the correlation between two random variables X and Y.
corr(X,Y) = cov(X,Y) / \sqrt(var(X)var(Y))
Define the conditional expectation for a random variable X conditional on Y in the case that they are discrete and continuous.
E[X|Y] = \int_(-\infty)^(\infty) dx x f_(X|Y) (x)
State the law of large numbers and the central limit theorem
Given a large enough sample size, the sample mean tends toward the population expectation.
CLT states that, in the limit N\to\infty, the sample mean is normally distributed, centred at the expectation and with vanishing variance
