Zermelo
Junior Member
- Joined
- Jan 7, 2021
- Messages
- 148
I'm following this playlist on probability theory
I have done a class on probability&stats and one in measure theory after that, so I never had a measure-theoretic start on probability theory.
I'm struggling on an exercise from this video (independence of random variables)
Independence is defined in the following way:
Let [imath](\Omega, A, P)[/imath] be a probability space and X: Omega -> R, Y: Omega -> R random variables.
X and Y are independent iff
[imath]X^{-1}((-\infty, x])[/imath] and [imath]Y^{-1}((-\infty, y])[/imath] are independent events in probability space [imath](\Omega, A, P)[/imath], for all real x and y.
From this definition we derive the condition for the CDFs:
[math]F_X(x)\cdot F_Y(y) = F_{(X, Y)}(x, y)[/math]
The exercise is:
Let [imath]\Omega = \Omega_1 \times \Omega_2[/imath], X and Y random variables on Omega, X(o1, o2) = f(o1), Y(o1, o2) = g(o2).
Prove that X and Y are independent.
This is my work:
[math]P(X^{-1}((-\infty, x]) \cap Y^{-1}((-\infty, y])) = \\ P(f^{-1}(\Omega_1) \times \Omega_2 \cap \Omega_1 \times g^{-1}(\Omega_2)) = \\ P(f^{-1}(\Omega_1) \times g^{-1}(\Omega_2))[/math]
I have to prove that [imath]P(X^{-1}((-\infty, x]) \cap Y^{-1}((-\infty, y])) = P(X^{-1}((-\infty, x])) \cdot P(Y^{-1}((-\infty, y]))[/imath]
And I'm a bit stuck.
First of all, product spaces come to mind, where from probability spaces (O1, A1, P1) and (O2, A2, P2) we can construct another probability space (O1xO2, ..., P1 x P2). So, in terms of my problem, this would mean that [imath] P(f^{-1}(\Omega_1) \times g^{-1}(\Omega_2)) = P_1(f^{-1}(\Omega_1)) \cdot P_2(g^{-1}(\Omega_2))[/imath]
BUT, I'm not sure if it goes the other way around. Ie, if O = O1xO2, that we can 'decompose' the probability measure P into two: P1 and P2. And I think that this isn't possible.
Is it possible? Am I missing something?
I have done a class on probability&stats and one in measure theory after that, so I never had a measure-theoretic start on probability theory.
I'm struggling on an exercise from this video (independence of random variables)
Independence is defined in the following way:
Let [imath](\Omega, A, P)[/imath] be a probability space and X: Omega -> R, Y: Omega -> R random variables.
X and Y are independent iff
[imath]X^{-1}((-\infty, x])[/imath] and [imath]Y^{-1}((-\infty, y])[/imath] are independent events in probability space [imath](\Omega, A, P)[/imath], for all real x and y.
From this definition we derive the condition for the CDFs:
[math]F_X(x)\cdot F_Y(y) = F_{(X, Y)}(x, y)[/math]
The exercise is:
Let [imath]\Omega = \Omega_1 \times \Omega_2[/imath], X and Y random variables on Omega, X(o1, o2) = f(o1), Y(o1, o2) = g(o2).
Prove that X and Y are independent.
This is my work:
[math]P(X^{-1}((-\infty, x]) \cap Y^{-1}((-\infty, y])) = \\ P(f^{-1}(\Omega_1) \times \Omega_2 \cap \Omega_1 \times g^{-1}(\Omega_2)) = \\ P(f^{-1}(\Omega_1) \times g^{-1}(\Omega_2))[/math]
I have to prove that [imath]P(X^{-1}((-\infty, x]) \cap Y^{-1}((-\infty, y])) = P(X^{-1}((-\infty, x])) \cdot P(Y^{-1}((-\infty, y]))[/imath]
And I'm a bit stuck.
First of all, product spaces come to mind, where from probability spaces (O1, A1, P1) and (O2, A2, P2) we can construct another probability space (O1xO2, ..., P1 x P2). So, in terms of my problem, this would mean that [imath] P(f^{-1}(\Omega_1) \times g^{-1}(\Omega_2)) = P_1(f^{-1}(\Omega_1)) \cdot P_2(g^{-1}(\Omega_2))[/imath]
BUT, I'm not sure if it goes the other way around. Ie, if O = O1xO2, that we can 'decompose' the probability measure P into two: P1 and P2. And I think that this isn't possible.
Is it possible? Am I missing something?
