Event independence little problem

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Homework Help Overview

The discussion revolves around the independence of two events defined within the unit square, specifically event A (y ≤ x) and event B (y ≤ 1 - x). The original poster also inquires about the independence of these events under the condition that B is restricted to x ≤ 1/4.

Discussion Character

  • Conceptual clarification, Assumption checking, Exploratory

Approaches and Questions Raised

  • Participants explore the mathematical definition of independence, questioning whether P(A|B) equals P(A). There is an attempt to express this relationship mathematically, but uncertainty remains about the correct formulation. The impact of the restriction B: x ≤ 1/4 on the independence of A and B is also discussed.

Discussion Status

Participants are engaged in exploring the implications of graphical representations of the events on their independence. Some guidance has been offered regarding the necessity of articulating what the graphs represent in terms of independence, indicating a productive direction in the discussion.

Contextual Notes

There is a noted challenge in expressing the dependence mathematically, particularly when considering the restriction of event B. The discussion reflects a mix of attempts to clarify definitions and the implications of graphical analysis.

atrus_ovis
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Homework Statement


0<=x<=1 0<=,y <=1
event A: y<=x
event B: y<=1-x
Are events A,B independent?
Also, if B: x<=1/4 , are A,B independent?

The Attempt at a Solution


if independent,
P(A|B) = P(A)
P(A|B) = P(A)P(B) / P(A)P(B)+(1-P(A))P(B)
... ?

For B: x<=1/4 they are obviously dependent, as B normalizes the event space to the part
where x<=1/4, thus affecting the probability of A, since it involves x.
I can't express it mathematically though, in the formula.
 
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atrus_ovis said:

Homework Statement


0<=x<=1 0<=,y <=1
event A: y<=x
event B: y<=1-x
Are events A,B independent?
Also, if B: x<=1/4 , are A,B independent?

The Attempt at a Solution


if independent,
P(A|B) = P(A)
P(A|B) = P(A)P(B) / P(A)P(B)+(1-P(A))P(B)
... ?

For B: x<=1/4 they are obviously dependent, as B normalizes the event space to the part
where x<=1/4, thus affecting the probability of A, since it involves x.
I can't express it mathematically though, in the formula.
Graph the inequalities y <= x and y <= 1 - x. Both of these are regions in the plane. The graphs should show you whether events A and B are independent.
 
Hm,okay.Thank you.
And is this considered a legitimate answer?
Graphs, i mean.
 
Well, the answer wouldn't be just a graph. You should say what it is that the graph represents as far as independence is concerned.
 
Yes, yes of course.
Thank you for your help Mark :)
 

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