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Notation of density functions

Marginal densities from a joint distributionTransference of properties from marginals to joint density functionsConditional density for $X$ given $X+Y=Z$X and Y are continuous random variables that are distributed jointly , and given $Xle Y$, then the following MUST occurConditional and Joint Density of Dependent Random VariablesProving normal distribution in Box MullerUniformly Distributed Marginal DensityIs there any reason not to use the notation $p_X mid Y = y(x)$?What is the conditional probability density $f_Y_1(x|y_1)$ if $y=G(x)$?fint joint and marginal distributions of two uniformy distributed variables over a specified region

0

$begingroup$

Assume two jointly normally distributed variables X,Y.

Then what I often see is that densities are written as follows:

$f_X(x)$ as the density of X, $f_Y(y)$ as the density of Y, $f_X,Y(x,y)$ as the joint density, $f_Y(x|y)$ as the conditional density, and so on.

My question: Why do people include this subindex? What is the meaning behind it? Does it add any information? E.g. why don't you just write $f(x)$ or $f(X)$ instead of $f_X(x)$? Does something like $f_Y(x)$ even make sense?

Thanks a lot!

probability probability-theory probability-distributions notation density-function

share|cite|improve this question

asked Mar 17 at 14:05

S_ZS_Z

121

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  To simply keep track of the random variable whose density is being considered. You pretty much answered this yourself by merely listing all the ways it can be used.
  $endgroup$
  – LoveTooNap29
  Mar 17 at 14:13
  

  
  
  
  


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  Of course the index can be left out. It is only there to make things more easy and to decrease the chance that our alphabet is not large enough. If you dislike them then feel free to do it without these indices, but do keep them properly apart of course. "Does something like $f_Y(x)$ even make sense?..." Well, can you give me any reason for doubting that? Btw, if $X$ is a random variable then also $f(X)$ is a random variable, and PDF's are not random variables.
  $endgroup$
  – drhab
  Mar 17 at 14:19
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Of course $f_Y(y)$ makes sense, but $f_Y(x)$?
  $endgroup$
  – S_Z
  Mar 17 at 14:22
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I made a typo (and repaired) in former comment. Again: what is your reason to doubt that $f_Y(x)$ makes sense? Personally I would not write the arguments if that is not necessary but would just do it with $f_X$ and $f_Y$. Further something like $int yf_Y(y)dy$ is exactly the same as $int xf_Y(x)dx$. Both denote the expectation of $Y$.
  $endgroup$
  – drhab
  Mar 17 at 14:25
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That helped, think I got it know. Thanks a lot!
  $endgroup$
  – S_Z
  Mar 17 at 14:32
  

  
  
  
  

add a comment | 

0

$begingroup$

Assume two jointly normally distributed variables X,Y.

Then what I often see is that densities are written as follows:

$f_X(x)$ as the density of X, $f_Y(y)$ as the density of Y, $f_X,Y(x,y)$ as the joint density, $f_Y(x|y)$ as the conditional density, and so on.

My question: Why do people include this subindex? What is the meaning behind it? Does it add any information? E.g. why don't you just write $f(x)$ or $f(X)$ instead of $f_X(x)$? Does something like $f_Y(x)$ even make sense?

Thanks a lot!

probability probability-theory probability-distributions notation density-function

share|cite|improve this question

asked Mar 17 at 14:05

S_ZS_Z

121

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  To simply keep track of the random variable whose density is being considered. You pretty much answered this yourself by merely listing all the ways it can be used.
  $endgroup$
  – LoveTooNap29
  Mar 17 at 14:13
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Of course the index can be left out. It is only there to make things more easy and to decrease the chance that our alphabet is not large enough. If you dislike them then feel free to do it without these indices, but do keep them properly apart of course. "Does something like $f_Y(x)$ even make sense?..." Well, can you give me any reason for doubting that? Btw, if $X$ is a random variable then also $f(X)$ is a random variable, and PDF's are not random variables.
  $endgroup$
  – drhab
  Mar 17 at 14:19
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Of course $f_Y(y)$ makes sense, but $f_Y(x)$?
  $endgroup$
  – S_Z
  Mar 17 at 14:22
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I made a typo (and repaired) in former comment. Again: what is your reason to doubt that $f_Y(x)$ makes sense? Personally I would not write the arguments if that is not necessary but would just do it with $f_X$ and $f_Y$. Further something like $int yf_Y(y)dy$ is exactly the same as $int xf_Y(x)dx$. Both denote the expectation of $Y$.
  $endgroup$
  – drhab
  Mar 17 at 14:25
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That helped, think I got it know. Thanks a lot!
  $endgroup$
  – S_Z
  Mar 17 at 14:32
  

  
  
  
  

add a comment | 

$begingroup$

Assume two jointly normally distributed variables X,Y.

Then what I often see is that densities are written as follows:

$f_X(x)$ as the density of X, $f_Y(y)$ as the density of Y, $f_X,Y(x,y)$ as the joint density, $f_Y(x|y)$ as the conditional density, and so on.

My question: Why do people include this subindex? What is the meaning behind it? Does it add any information? E.g. why don't you just write $f(x)$ or $f(X)$ instead of $f_X(x)$? Does something like $f_Y(x)$ even make sense?

Thanks a lot!

probability probability-theory probability-distributions notation density-function

share|cite|improve this question

asked Mar 17 at 14:05

S_ZS_Z

121

$endgroup$

share|cite|improve this question

asked Mar 17 at 14:05

S_ZS_Z

121

share|cite|improve this question

asked Mar 17 at 14:05

S_ZS_Z

121

asked Mar 17 at 14:05

S_ZS_Z

121

asked Mar 17 at 14:05

S_ZS_Z

121