1) Random Variables Flashcards
What is a sample space
The set of possible outcomes , Ω
What is an event
A susbet of the sample space
Describe some of the common notation used in probability
What does it mean if A and B are disjoint
A ∩ B = ∅
What is the Probability Space
Conatins 3 elements (Ω, F, P) -
* Ω - sample space
* F - subsets of Ω
* P - probability measure
When is a collection F of subsets of Ω a σ-algebra
What is a probability measure P on (Ω, F)
A function P: F → [0, 1] satisfying:
(i) P(Ω) = 1,
(ii) P(∅) = 0
When is a function X : Ω → R measurable
If the sets { ω | X(ω) ⩽ x } ∈ F for all x ∈ R
What is a random variable
A measurable function X : Ω → R
What is the range of X
The set X(Ω) of all possible values of X
What is the cumulative distribution function (cdf)
What are the properties of the probability of a cdf
What are the properties of the cdf
What is a discrete random variable
If the range of the random variable X can only assume countably many values, RX = {x1, x2, … }
What is the probability mass function (pmf)
What are the properties of the pmf
What is the Bernoulli Distribution
What is the Binomial Distribution
What is the Geometric Distribution
What is the Poisson Distribution
What is the uniform distribution (discrete)
What is the cdf of discrete random variables
The c.d.f. of a discrete random variable is a step function
What is the probability density function (pdf)
The function fX : R → [0, ∞) such that
What is the relationship between pdf and cdf
What are the properties of the pdf
What is the uniform distribution (continuous)
What is the exponential distribution
What is the normal distribution
What is the pdf of an induced random variable Y = g(X)
Suppose that g is either strictly increasing or decreasing
Describe the proof of the pdf of an induced random variable Y = g(X)
