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What is the conditional density function of $Z|Z = X$, where $Z = max{X, Y}$, X~exp($lambda_1$), Y~exp($lambda_2$) and X, Y are independent?

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1

$begingroup$

Given that $X $~$ $exp$(lambda_1)$, $Y $~$ $exp$(lambda_2)$, and $X$,$Y$ are independt.

$Z$=$max{X,Y}$

Find the conditional density function of $Z$ given $Z$=$X$.

I get pdf of $Z$ is $lambda_1 e^{-lambda_1z} (1 - e^{-lambda_2z})+lambda_2 e^{-lambda_2z} (1 - e^{-lambda_1z})$

But how to write the conditional density function of $Z|Z=X$?

probability-distributions stochastic-processes random-variables random

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edited 17 hours ago

ChengBang Chen

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ChengBang ChenChengBang Chen

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ChengBang Chen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  $begingroup$
  Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
  $endgroup$
  – dantopa
  yesterday
  

  
  
  
  

add a comment | 

1

$begingroup$

Given that $X $~$ $exp$(lambda_1)$, $Y $~$ $exp$(lambda_2)$, and $X$,$Y$ are independt.

$Z$=$max{X,Y}$

Find the conditional density function of $Z$ given $Z$=$X$.

I get pdf of $Z$ is $lambda_1 e^{-lambda_1z} (1 - e^{-lambda_2z})+lambda_2 e^{-lambda_2z} (1 - e^{-lambda_1z})$

But how to write the conditional density function of $Z|Z=X$?

probability-distributions stochastic-processes random-variables random

share|cite|improve this question

edited 17 hours ago

ChengBang Chen

asked yesterday

ChengBang ChenChengBang Chen

62

New contributor

ChengBang Chen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
  $endgroup$
  – dantopa
  yesterday
  

  
  
  
  

add a comment | 

$begingroup$

Given that $X $~$ $exp$(lambda_1)$, $Y $~$ $exp$(lambda_2)$, and $X$,$Y$ are independt.

$Z$=$max{X,Y}$

Find the conditional density function of $Z$ given $Z$=$X$.

I get pdf of $Z$ is $lambda_1 e^{-lambda_1z} (1 - e^{-lambda_2z})+lambda_2 e^{-lambda_2z} (1 - e^{-lambda_1z})$

But how to write the conditional density function of $Z|Z=X$?

probability-distributions stochastic-processes random-variables random

share|cite|improve this question

edited 17 hours ago

ChengBang Chen

asked yesterday

ChengBang ChenChengBang Chen

62

New contributor

ChengBang Chen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited 17 hours ago

ChengBang Chen

asked yesterday

ChengBang ChenChengBang Chen

62

New contributor

ChengBang Chen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited 17 hours ago

ChengBang Chen

asked yesterday

ChengBang ChenChengBang Chen

62

New contributor

ChengBang Chen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked yesterday

ChengBang ChenChengBang Chen

62

New contributor

ChengBang Chen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked yesterday

ChengBang ChenChengBang Chen

62