# Proabibility - Random variables independence question

• serhannn
In summary, the question is whether X and Y are independent, given that their joint density is constant in a circular region and zero outside. To determine independence, the multiplication of the marginal densities must equal the joint density. However, the integration limits in a circular region are not as straightforward as in simple boundaries, and it may be helpful to convert to polar coordinates first. Being independent means that knowing the value of one variable does not increase knowledge of the other variable.
serhannn

## Homework Statement

Two variables, X and Y have a joint density f(x,y) which is constant (1/∏) in the circular region x2+y2 <= 1 and is zero outside that region
The question is: Are X and Y independent?

## Homework Equations

Well, I know that for two random variable to be independent, multiplication of their marginal densities must equal their joint denstiy, i.e.
f(x,y)=fX(x)*fY(y)

## The Attempt at a Solution

My problem is I am confused about how to select the integration limits. I know how to do it when simple boundaries are given (like x<2 and y>1, etc.) but within a circular region, I just could not figuer out how to do it. Should I, for example, integrate y from -sqrt(1-x2 ) to sqrt(1-x2) or is that a wrong approach? How can I select the integration limits in a circular region? Is it a better approach to convert to polar coordinates first and then integrate?

Thanks a lot for your help.

To be independent means that if I give you value of x, that doesn't increase your knowledge on value of y. If I tell you that x=1, can you tell me anything about y?

## 1. What is probability?

Probability is the measure of the likelihood that an event will occur. It is typically expressed as a number between 0 and 1, with 0 representing impossibility and 1 representing certainty.

## 2. What are random variables?

Random variables are numerical values that are assigned to the outcomes of a random process or experiment. They can have various probability distributions and are used to model uncertain events.

## 3. What is independence in terms of probability and random variables?

Independence refers to the lack of relationship between two or more events or variables. In terms of probability and random variables, it means that the occurrence or value of one does not affect the probability or value of the other.

## 4. How do you determine if two random variables are independent?

In order to determine if two random variables are independent, you can use the definition of independence which states that the joint probability of two events occurring is equal to the product of their individual probabilities. If this condition is met, then the variables are considered independent.

## 5. Can two events be independent but have a correlation?

Yes, it is possible for two events to be independent but still have a correlation. This is because correlation measures the strength of the relationship between two variables, while independence measures the lack of relationship. So, even if the variables are not related, they may still have a correlation due to chance.

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