Stats 214 Chapter 2 NOTES

CH2 Discrete Probability Distributions
Axioms of Probability:




o Based on these axioms, the following formulas can be derived:




o Note the formulation of a conditional probability:




A random variable X is a mapping from sample space (Ω) to the real line (R).
o X:Ω→R


The support of X (x) is the smallest closed set for which P(X ∈ x) = 1

o It is the set of values that X can assume.

The form of x determines the type of probability distribution we are dealing
with:
o Discrete Probability Distribution = x is a discrete set such as {0, 1, 2}
(finite countable) or {0, 1, 2, . . .} (finite uncountable).
o Continuous Probability Distribution = x is a continuous set (usually
intervals such as [0, 1]).

, Univariate Discrete Distributions
o a discrete random variable X is a random variable for which x is a
discrete set.
Mass & Distribution Functions
The probability mass function (pmf) of X is a function pX : x → [0,1]



o Where, pX(x) ≥ 0 and Σ[pX(x)] = 1



The distribution function of X is a function FX : R → [0, 1]




Expected Values & Variances
The expected value of X is defined as




The variance of X is defined as




Let a, b & c be constants → the following holds: