Minimum of random and constant scaled with random variable

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Suppose that $X$ and $Y$are independent random variables, while $c geq 0$ is a constant. More specifically, $X$ is the $n$-fold convolution of $Y$. Is it then correct to say the following?

$min(Y,c)+X = min(Y+X, c+X)$

Obviously this would be correct for only constants. But does it hold for random variables?

random-variables maxima-minima

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edited Jul 20 at 9:54

asked Jul 20 at 9:40

JoeS

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up vote
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Suppose that $X$ and $Y$are independent random variables, while $c geq 0$ is a constant. More specifically, $X$ is the $n$-fold convolution of $Y$. Is it then correct to say the following?

$min(Y,c)+X = min(Y+X, c+X)$

Obviously this would be correct for only constants. But does it hold for random variables?

random-variables maxima-minima

share|cite|improve this question

edited Jul 20 at 9:54

asked Jul 20 at 9:40

JoeS

62

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Suppose that $X$ and $Y$are independent random variables, while $c geq 0$ is a constant. More specifically, $X$ is the $n$-fold convolution of $Y$. Is it then correct to say the following?

$min(Y,c)+X = min(Y+X, c+X)$

Obviously this would be correct for only constants. But does it hold for random variables?

random-variables maxima-minima

share|cite|improve this question

edited Jul 20 at 9:54

asked Jul 20 at 9:40

JoeS

62

Suppose that $X$ and $Y$are independent random variables, while $c geq 0$ is a constant. More specifically, $X$ is the $n$-fold convolution of $Y$. Is it then correct to say the following?

$min(Y,c)+X = min(Y+X, c+X)$

Obviously this would be correct for only constants. But does it hold for random variables?

random-variables maxima-minima

share|cite|improve this question

edited Jul 20 at 9:54

asked Jul 20 at 9:40

JoeS

62

share|cite|improve this question

share|cite|improve this question

edited Jul 20 at 9:54

edited Jul 20 at 9:54

edited Jul 20 at 9:54

asked Jul 20 at 9:40

JoeS

62

asked Jul 20 at 9:40

JoeS

62

asked Jul 20 at 9:40

JoeS

62

62

add a comment | 

add a comment | 

1 Answer
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This has nothing to do with random variables. For any three real numbers $a,b,c$ we have $min (a,c)+b =min (a+b,c+b)$ so the same holds for random variables. [You can just verify the equation for the cases $a leq c$ and $a>c$].

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answered Jul 20 at 9:55

Kavi Rama Murthy

20.6k2830

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the quick reply!
  – JoeS
  Jul 20 at 10:07
  

  
  
  
  

add a comment | 

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1 Answer
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1 Answer
1

active

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active

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oldest

votes

up vote
1
down vote

This has nothing to do with random variables. For any three real numbers $a,b,c$ we have $min (a,c)+b =min (a+b,c+b)$ so the same holds for random variables. [You can just verify the equation for the cases $a leq c$ and $a>c$].

share|cite|improve this answer

answered Jul 20 at 9:55

Kavi Rama Murthy

20.6k2830

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the quick reply!
  – JoeS
  Jul 20 at 10:07
  

  
  
  
  

add a comment | 

up vote
1
down vote

This has nothing to do with random variables. For any three real numbers $a,b,c$ we have $min (a,c)+b =min (a+b,c+b)$ so the same holds for random variables. [You can just verify the equation for the cases $a leq c$ and $a>c$].

share|cite|improve this answer

answered Jul 20 at 9:55

Kavi Rama Murthy

20.6k2830

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the quick reply!
  – JoeS
  Jul 20 at 10:07
  

  
  
  
  

add a comment | 

up vote
1
down vote

up vote
1
down vote

This has nothing to do with random variables. For any three real numbers $a,b,c$ we have $min (a,c)+b =min (a+b,c+b)$ so the same holds for random variables. [You can just verify the equation for the cases $a leq c$ and $a>c$].

share|cite|improve this answer

answered Jul 20 at 9:55

Kavi Rama Murthy

20.6k2830

This has nothing to do with random variables. For any three real numbers $a,b,c$ we have $min (a,c)+b =min (a+b,c+b)$ so the same holds for random variables. [You can just verify the equation for the cases $a leq c$ and $a>c$].

share|cite|improve this answer

answered Jul 20 at 9:55

Kavi Rama Murthy

20.6k2830

share|cite|improve this answer

share|cite|improve this answer

answered Jul 20 at 9:55

Kavi Rama Murthy

20.6k2830

answered Jul 20 at 9:55

Kavi Rama Murthy

20.6k2830

answered Jul 20 at 9:55

Kavi Rama Murthy

20.6k2830

20.6k2830

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the quick reply!
  – JoeS
  Jul 20 at 10:07
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks for the quick reply!
  – JoeS
  Jul 20 at 10:07
  

  
  
  
  

Thanks for the quick reply!
– JoeS
Jul 20 at 10:07

Thanks for the quick reply!
– JoeS
Jul 20 at 10:07

add a comment | 

 

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