Probability Flashcards
What is a finite sample space?
Finite sample space has a finite number of outcomes/elements.
What is a countably infinite sample space?
Countably infinite sample space is a sample space which outecomes can be put into to a one-to-one correspondence with the set of natural numbers.
What is a discrete sample space?
If a sample space is finite or countably infinite it can also be called discrete.
Another name for a discrete sample space.
Countable
What is a continuous sample space?
Continuous sample space contains outcomes that can take any value of any real number on a certain interval.
What are de Morgans laws?
(A ∩ B)^c = A^c U B^c.
(A U B)^c = A^c ∩ B^c.
What is a sure event?
Set S - which is the whole sample space.
What is a null event?
Empty set
What is an elementary event?
One outcome in the sample space/experiment.
What are mutually exclusive events? (set notation)
A∩ B = ∅
What is a relative frequency?
If n(x) represents the number of times event x has occurred in the total number of experiment trials N, then fx= n(x) / N
What is statistical regularity?
P(x) = lim fx = lim n(x) / N (n is goingt to infinity)
What is the primary objective of probability modelling?
Given a sample space S, we aim to assign a real number P(A) to each possible outcome of A, which describes the likelihood that A will occur if the experiment is performed.
What is an event?
An event is a subset of a sample space.
How to think of P(A) as a function? What is the domain? Range?
All possible outcomes are the domain and the probabilities are the range.
What are three axioms of probability?
- 0<=P(A)<=1
- P(S) = 1
- P ( U Ai) =P(A1)UP(A2)U…UP(Ai) = P(A1)+P(A2)+…P(Ai)
If A1, A2 are pairwise mutually exclusive events.
What is a classical probability model?
Implies that each elementary event has equal probability
What is an object chose of random?
If an object is chosen at random it means it had the same probability of being picked as all the other object in the sample space.
P(A U B U C) ?
P( A ) + P (B) + P(C) - P(A ∩ B)- P(A ∩ C) - P(C ∩ B) + P (A ∩ B ∩ C)
If A is a subset of B, then P(A) ? P(B)
< / > / =
<=
What is the formula for conditional probability? Explain
P(A|B) = P(A ∩ B) / P(B)
Conditional probability determines a probability of an event on the reduced sample space.
P(A|B) determines how likely event A, given that B has occurred. As usual, we can look what is the total number of events when A and B have occurred and divide it by the total number of events B. Dividing the numerator and denominator by the total sample size allows determining the conditional prob by unconditional prob.
What is the total law of probability?
P(A) = P(A1|B)P(B) + P(A2|B)P(B) + … + P(An|B)P(B)
What is the Multiplication law of probability?
P(A ∩ B) = P(A|B)P(B) = P(B|A)P(A)
What is a marginal probability?
Basically anytime you are in interested in a single event irrespective of any other event (i.e. “marginalizing the other event”), then it is a marginal probability.
Which two events are called independent?
A and B are independent if P(A ∩ B) = P(A) * P(B)
What is Bayes’ Rule?
P(B|A) = P(A|B)*P(B) / P(A)
Which k events are independent / mutually independent?
P(A1 ∩ A2 ∩ An) = P(A1) * P(A2)*P(An)
What is a random variable?
Random variable is a function defined over the sample space S, that assigns a real number X(e) = x
What is a discrete random variable?
If the set of all possible values of a random var X is a countable set, then X is a discrete random variables.
What is the probability density function of a discrete random variable?
PDF is f(x) = P [ X = x]
It assigns the probability of each possible value x.
A function is a pdf if and only if f(xi)>=0 and sum over all x f(xi) = 1
What is another name of pdf?
Probability mass function
What is a cumulative distribution function?
F(x) = P [X<=x]
A function is a cdf if and only if what? (4)
lim x -> - inf F(x) = 0
lim x ->1 F(x) = 1
lim h -> 0+ F(x+h) = F(x) (right continious)
a<b> F(a) <= F(b)</b>
What is the general relationship between F(x) and f(x)?
f(x1) = F ( x1 ) f(xi) = F ( xi ) - F ( xi-1 )
What is the expected value of a discrete random variable X?
If pdf f(x), then E ( X) = (Sum over all xi) f(xi)*xi
What are other names for the expected value?
Mean
Expectation
What is a continuous random variable?
Variable can be considered continious if there is a pdf function of X, such that the CDF is
F(x) = x to (-inf) ∫ f(t)dt
With a continuous random variable how to get from CDF to pdf?
f(x) = F’(x)
With a continuous random variable how to get from pdf to CDF?
F(x) = x to (-inf) ∫ f(t)dt
Consider events of a continuous random variable of the form [X ∈ I], where I is an interval. What is important to remember?
The probability of the event is the same weather I include the endpoints or not.
When is a function f(x) a pdf for some continuous random variable X?
If and only if
f(x) > = 0
and for all real x:
(inf) to (-inf) ∫ f(x)dx = 1
How to find a P[a
P[a
What is the expected value of a continuous random variable?
E(X) = (inf) to (-inf) ∫ x * f(x)dx
What are other notations for E(X)?
μ , μx,
Mean / expectation
Center of mass
What does a percentile p of a distribution of a continuous random variable do?
It indicates the value below which a given percentage of observations in a group of observations fall.
F(x) = p
What is a median of a distribution?
50th percentile
What is a mode of a distribution?
If there is a uniques maximum of pdf at x = m0, such as max f(x) = f( m0), then m0 is a mode.
When is a distribution symmetric about c?
Distribution symmetric about c if f(c-x) = f(c+x) for all x.
What is another name for a skewed distribution?
Asymmetric.
Formula for the variance of X
var(x) = E[ (X-μ )^2 ]
or
E[ (X^2)] +E(X)^2 = E[ (X^2)] + μ^2
A formula for the variance of X (2)
var(x) = E[ (X-μ )^2 ]
or
E[ (X^2)] +E(X)^2 = E[ (X^2)] + μ^2
Kth moment about the mean
μk = E(X-μ)^k
Var(aX+b) = ?
a^2 * Var(X)
What can you say about the third moment about mean of X if the distribution is symmetric?
μ3 = 0
What is Markov inequality?
What is the proof for Markov inequality?
P (|X|> = c) = E[X] / c
Proof: see page 76
What is the Chebishev inequality?
P(|X - μ| >= kσ) <= 1 / K^2
and
P(|X - μ| <= kσ) >=1 - 1 / K^2
What is the Chebishev inequality?
P(|X - μ| >= kσ) <= 1 / K^2
and
P(|X - μ| <= kσ) >=1 - 1 / K^2
Degenerate distribution
Distribution with σ^2= 0 and distribution . concentrated at one point.
What is a moment of a function?
The moments are special expected values, which include the mean and variance as particular cases, and also provide descriptive measures for other characteristics
such as skewness of a distribution.
