Let $(Ω,F,P)$ be a probability space. Suppose $X(ω)=C, ∀ ω∈Ω$. Show that $X$ is independent of any other random variable.

Question

Let $(\Omega, F, P)$ be a probability space. Suppose $X(\omega)=C$, $\forall\omega \in \Omega$ where $C$ is a constant.

How do I show that $X$ is independent of any other random variable?

I noticed that you have just joined MSE today. Welcome! You are expected to show some effort in answering the question yourself before posting the question. We don't want do your homework problems here. — Kavi Rama Murthy, Mar 06 '20 at 10:01

score 2 · Answer 1 · answered Mar 06 '20 at 09:57

2

Hint: $X^{-1}(A)=\emptyset$ or $X^{-1}(A)=\Omega$ for any Borel set $A$. Now apply the definition.

answered Mar 06 '20 at 09:57

Kavi Rama Murthy

311,013

score 0 · Answer 2 · answered Mar 06 '20 at 10:41

From measure theory point of view, two Random Variables are independent iff for all measurable sets in their sigma algebra they are independent (i.e. standard definition with pushing product outside the probability)

As Kavi gave you a hint: try to think what is the result of applying preimage of $\cal{X}$ for different events (from sigma algebra). Well, either it includes that constant $C$ or it does not. Therefore, it is either empty (probability is 0) or everything (probability is 1). So on your RHS (the one with product of probabilities) you have either 0 or you forget about $\cal{X}$ random variable (as multiplying by 1 does not change anything). Now, let's inspect LHS (probability of two events happening at the same time) and you quickly notice that it is either 0 or probability of the second event and the cases are disjoint.

Let $(Ω,F,P)$ be a probability space. Suppose $X(ω)=C, ∀ ω∈Ω$. Show that $X$ is independent of any other random variable.

2 Answers2