Lecture 16

A probability space is a triplet (Ω, A, P). What does each component represent?

• Ω is the set of all possible outcomes (the 'sample space').

• A is the set of all elementary events, like {1}, {2}, etc.

• A is the σ-algebra, which is the set of 'events' (measurable subsets of Ω).

Consider the discrete measure defined on Ω = {x₁, x₂, …} with probabilities P(xᵢ) = pᵢ. Which of the following is/are correct?

• Each pᵢ can be any non-negative number.

• Each pᵢ must be between 0 and 1 (inclusive).

• The sum of all pᵢ must be 1.

• pᵢ can be negative if other probabilities compensate to sum to 1.

Tags

CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

What are the defining properties of a probability measure, P, on a measurable space (Ω, A)?

• The elements of A are called events.

• It is countably additive.

• P(A) is the number of elements in A divided by the total number of elements in Ω.

• P(A) = 1 if A is not empty and 0 otherwise.

Which of the following best describes the Dirac measure δₓ at a point x in ℝ?

• δₓ(A) = 1 if x ∈ A, and 0 otherwise.

• δₓ(A) = |A − x|.

• δₓ(A) = 0 for all A, because it is a degenerate measure.

• δₓ(A) = the length of A when x ∈ A.

Which of the following statements about the normal distribution N(μ, σ²) on ℝ is correct?

• It is a discrete measure with finite support.

• It is absolutely continuous with respect to the Lebesgue measure.

•  It can be written as δₓ for some x in ℝ.

• It's probability density function is f(x) = (1 / √(2πσ²)) exp(−(x − μ)² / (2σ²)).

Tags

CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

Which of the following are examples of discrete probability measures?

Tags

CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

• Every μ-null set is also a γ-null set.

• Every γ-null set is also a μ-null set.