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:
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:
