Are Random Variables X and Y Related If Their Conditional Probabilities Change?

In summary, the conversation discusses the relationship between two random variables X and Y. It is stated that X and Y are related if the joint probability density function changes as Y changes. This is shown through various equations and examples. The conversation also mentions that if X and Y are not related, then they are independent. This means that the probability of X being less than a certain value and Y being less than another value is equal to the product of their individual probabilities. The relationship between X and Y is symmetric, meaning it holds true for both variables.
  • #1
Kuma
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1) I'm trying to prove that two R.V.s X & Y are related iff Y & X are related. Assuming they are discretely distributed.

So basically from what I've learned is that two R.V.s are related if the joint pdf changes as Y changes. So basically if f(X|Y=yi) changes when i changes. So from that definition this is what I came up with.

if I have 2 pdf's and assuming X and Y are related then

f(X=x1|Y=y1) = P(X=x1 n Y=y1)/P(Y=y1)

should not be the same as:

f(X=x1|Y=y2) = P(X=x1 n Y=y2)/P(Y=y2)

However if Y and X are not related then:

f(Y=y1|X=x1) = P(Y=y1 n X=x1)/P(X=x1)

should be the same as:

f(Y=y1|X=x2) = P(Y=y1 n X=x2)/P(X=x2)

But P(X=x1 n Y=y1) = P(Y=y1 n X=x1) thus:

f(Y=y1|X=x1)*P(X=x1)/P(Y=y1) = f(X=x1|Y=y1)

So we can see that f(X=x1|Y=y1) depends on f(Y=y1|X=x1), so if X and Y are related, it should mean that Y and X are related as well? Since f(Y=y1|X=x1) = f(Y=y1|X=x2), I can put in any f(Y|X=xi) in there, and the left side should remain unchanged, but it is contradictory to the right side since it has to change if X and Y are related. I'm not sure if that's the right way to prove it.
 
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  • #2
Random variables X and Y are independent means P(X<x and Y<y) = P(X<x)P(Y<y) for all x and y. If this relationship does not hold, they are dependent (you call related). The equation is symmetric in X and Y.
 

FAQ: Are Random Variables X and Y Related If Their Conditional Probabilities Change?

1. What is a simple conditional proof?

A simple conditional proof is a logical proof technique used in propositional logic to establish the validity of a conditional statement. It involves assuming the antecedent of the conditional statement and using logical rules and previously established premises to prove the consequent.

2. When is a simple conditional proof useful?

A simple conditional proof is useful when trying to prove a conditional statement, as it allows for a straightforward and systematic approach to proving the statement's validity. It is also used in many other fields, such as mathematics and computer science, to construct rigorous arguments.

3. What are the steps involved in a simple conditional proof?

The steps involved in a simple conditional proof are as follows: 1. Begin by assuming the antecedent of the conditional statement. 2. Use logical rules, such as Modus Ponens or Modus Tollens, and previously established premises to derive the consequent. 3. Once the consequent has been derived, the proof is complete and the conditional statement is proven to be valid.

4. What are the benefits of using a simple conditional proof?

There are several benefits to using a simple conditional proof, including: 1. It provides a clear and systematic approach to proving conditional statements. 2. It allows for the use of logical rules to derive the consequent, making the proof more rigorous. 3. It can be applied to a wide range of fields and topics, making it a versatile proof technique.

5. Are there any limitations to using a simple conditional proof?

While simple conditional proof is a useful technique, it does have limitations. These include: 1. It can only be used to prove conditional statements, and not other types of logical statements. 2. It relies on the assumption of the antecedent, which may not always be a valid assumption. 3. It may be more complex to use in more complicated proofs compared to other proof techniques.

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