Can Z and U be considered independent in this scenario?

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The discussion revolves around proving the independence of random variables Z and U, defined in relation to two exponential random variables X and Y with different rates. Z is a discrete variable representing whether X is less than or equal to Y, while U is defined as the minimum of X and Y. A participant suggests deriving the cumulative distribution function (CDF) of Z instead of the probability density function (PDF), as Z is discrete and lacks a PDF. The key approach involves demonstrating that the joint CDF of Z and U equals the product of their marginal CDFs. This method provides a pathway to establish their independence.
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 ?:rolleyes:

lolypop
 
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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.
 

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