Can Z and U be considered independent in this scenario?

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In summary, the conversation discusses a question on proving the independence of two new random variables, Z and U, defined in terms of two exponential random variables, X and Y. The conversation mentions trying to derive the pdf of Z but getting stuck, and someone suggests considering the joint cdf instead.
  • #1
lolypop
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Hi ,
I got this question in my midterm today but up till now I don't know how to solve it ,

The Question is as follow :
If X and Y are two exponential Rv with different lambda . and there's a new Rvs Z and U are defined such that :

Z= 0 : X<Y and 1 : X=> Y

U= min(X,Y)

and the Question asked to proof that Z and U are independent .

So I started my solution by deriving the pdf of U since I know how to then tried to derive the pdf of Z but didn't know where to start and got stuck . :frown:

Can anyone tell me of a way to derive the pdf of Z . or is there another way to solve the problem ?:uhh:

lolypop
 
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  • #2
Z is discrete and doesn't have a pdf - one way around is to consider the cdf instead, i.e. show that the joint cdf of Z and U is a product of the marginal cdfs.
 

1. What is a jointly distributed random variable?

A jointly distributed random variable is a type of random variable that has multiple dimensions or components, each of which can take on different values. It describes the relationship between two or more random variables and how they are related to each other.

2. How is a jointly distributed random variable different from a single random variable?

A single random variable has only one dimension or component, while a jointly distributed random variable has multiple dimensions or components. This means that a joint distribution takes into account the relationship between two or more random variables, while a single random variable only describes the characteristics of one variable.

3. What is the purpose of using a joint distribution when analyzing data?

A joint distribution allows for the analysis of the relationship between two or more random variables, which can provide a deeper understanding of the data. It can also help identify patterns or correlations between variables that may not be apparent when looking at them individually.

4. How is a joint distribution represented in mathematical notation?

A joint distribution is typically represented using the notation P(X,Y), where X and Y are the two random variables being analyzed. This notation shows the probability of both X and Y occurring simultaneously.

5. What is the difference between a joint distribution and a conditional distribution?

A joint distribution describes the relationship between two or more random variables, while a conditional distribution describes the probability of one random variable given the occurrence of another. In other words, a conditional distribution takes into account a specific condition, while a joint distribution looks at the overall relationship between variables.

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