independence of random variables (Aj = {-1, 1} for all j)

[tapi]

I want to show the independence of a finite collection of random variable using this definition: A finite collection of random variables X1, X2, . . . , Xn is mutually independent if the sets (Xj ∈ Aj ) are mutually independent for all events Aj in the ranges of the corresponding Xj.
Here, Xi's are steps in 1D of a random walk, i.e., P(X_i=1)=P(X_i=-1)=1/2 and I know that P(X_k1=x_k1, X_k2=x_k2,..., X_kn=x_kn)=2^{-n}.

Now Aj={-1,1} for all j, but I can't figure out what to do next? How's $X_j \in A_j$ a set? Do the events refer to $X_j=\{1\},\{-1\}$?

 

[blamocur]

I want to show the independence of a finite collection of random variable using this definition: A finite collection of random variables X1, X2, . . . , Xn is mutually independent if the sets (Xj ∈ Aj ) are mutually independent for all events Aj in the ranges of the corresponding Xj.
Here, Xi's are steps in 1D of a random walk, i.e., P(X_i=1)=P(X_i=-1)=1/2 and I know that P(X_k1=x_k1, X_k2=x_k2,..., X_kn=x_kn)=2^{-n}.

Now Aj={-1,1} for all j, but I can't figure out what to do next? How's $X_j \in A_j$ a set? Do the events refer to $X_j=\{1\},\{-1\}$?

The definition I am familiar with says that [imath]A_j[/imath] is a set and [imath]X_j \in A_j[/imath] defines an event.

What does it mean to say that, for example, [imath]X_1[/imath] and [imath]X_2[/imath] are independent?

 

[tapi]

The definition I am familiar with says that [imath]A_j[/imath] is a set and [imath]X_j \in A_j[/imath] defines an event.

What does it mean to say that, for example, [imath]X_1[/imath] and [imath]X_2[/imath] are independent?

That is what I'm struggling with - to rephrase a problem in order to fit this definition. Could you please help me formulate it terms of Aj's? I suppose Aj is {-1,1} and Xj belongs to Aj could be restated as xj belongs to {\phi, {1},{-1},{1,-1}}?

 

[blamocur]

That is what I'm struggling with - to rephrase a problem in order to fit this definition. Could you please help me formulate it terms of Aj's? I suppose Aj is {-1,1} and Xj belongs to Aj could be restated as xj belongs to {\phi, {1},{-1},{1,-1}}?

Which definition of random variable are you using?
Have you looked at the Wikipedia page I've linked to?