3) Random Variables

Question 1

What is a measurable space

Answer 1

Question 2

What is a measurable function

Answer 2

Question 3

What are the conditions for a function f:Ω→S to be measurable in the context of measurable spaces (Ω,F) and (S,B)

Answer 3

Question 4

What is a Random Variable

Answer 4

A random variable is a measurable function X:Ω→R where (Ω,F,P) is a probability space, and (R,B(R)) is the measurable space of real numbers with the Borel σ-algebra B(R)

Question 5

What is a Borel Function

Answer 5

A Borel function is a measurable function f:R→R where both the domain (R,B(R)) and the codomain (R,B(R)) are the real numbers equipped with the Borel σ-algebra B(R)

Question 6

How do definitions of a random variable and a Borel function extend to R^n for n≥2

Answer 6

Question 7

In the context of measurable spaces, what can be said about the composition of two measurable functions

Answer 7

Question 8

If X is a random variable and f is a Borel function, what can be said about the composition f∘X

Answer 8

f ◦ X is a random variable

Question 9

What can be said about the measurability of continuous functions from R→R

Answer 9

If f:R→R is a continuous function, then f is measurable. This means that every continuous function on R is also a Borel function

Question 10

If f and g are measurable functions from Ω→R, what can be said about a linear combination of these functions

Answer 10

If f:Ω→R and g:Ω→R are measurable functions, α,β∈R are fixed constants, then the linear combination αf+βg:Ω→R is also measurable

Question 11

If X and Y are random variables, and α,β∈R are fixed constants, what can be said about αX+βY

Answer 11

If X and Y are random variables, then the linear combination αX+βY is also a random variable

Question 12

If (Ω,F) is a measurable space and fn :Ω→R are measurable functions, which limits and operations on fn are also measurable

Answer 12

If Xn are random variables, then the following are also random variable

Question 13

What is the law of a random variable X on a probability space (Ω,F,P)

Answer 13

Question 14

What is the distribution function FX of a random variable X, and what are its key properties

Answer 14

Question 15

What is the density function f X of a random variable X with an absolutely continuous distribution function F X

Answer 15

Question 16

What is the σ -algebra generated by a random variable X

Answer 16

Question 17

When is a random variable X σ(Y)-measurable for another random variable Y

Answer 17

A random variable X is σ(Y)-measurable if and only if there exists a Borel function f:R→R such that X=f(Y).
This means X can be expressed as a function of Y, ensuring X is measurable with respect to the σ-algebra generated by Y

Question 18

What is the σ-algebra generated by a family of random variables Xi for i∈I

Answer 18