Generalised (general) Uniform Distribution (continuous)











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I have seen the Uniform Distribution/a uniform random variable for some interval in $mathbb{R}$. For example $U(a,b)$ has probability density function $frac{1}{b-a}$ (noting this is the 'volume' of the interval)



My question : Is there a general Uniform Distribution/a general uniform random variable for a set $Omega$ which is a subset of $mathbb{R}^{n}$ , I assume if so the p.d.f will be the volume of $Omega$ ?



If this exists could anyone tell me some of its properties like its mean, its variance, its p.d.f, /any other interesting things about it?



If this distribution (or random variable) has not been extensively studied and the above properties can not be provided, then can we make $Omega$ 'nice enough' so that it does?










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    up vote
    0
    down vote

    favorite












    I have seen the Uniform Distribution/a uniform random variable for some interval in $mathbb{R}$. For example $U(a,b)$ has probability density function $frac{1}{b-a}$ (noting this is the 'volume' of the interval)



    My question : Is there a general Uniform Distribution/a general uniform random variable for a set $Omega$ which is a subset of $mathbb{R}^{n}$ , I assume if so the p.d.f will be the volume of $Omega$ ?



    If this exists could anyone tell me some of its properties like its mean, its variance, its p.d.f, /any other interesting things about it?



    If this distribution (or random variable) has not been extensively studied and the above properties can not be provided, then can we make $Omega$ 'nice enough' so that it does?










    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I have seen the Uniform Distribution/a uniform random variable for some interval in $mathbb{R}$. For example $U(a,b)$ has probability density function $frac{1}{b-a}$ (noting this is the 'volume' of the interval)



      My question : Is there a general Uniform Distribution/a general uniform random variable for a set $Omega$ which is a subset of $mathbb{R}^{n}$ , I assume if so the p.d.f will be the volume of $Omega$ ?



      If this exists could anyone tell me some of its properties like its mean, its variance, its p.d.f, /any other interesting things about it?



      If this distribution (or random variable) has not been extensively studied and the above properties can not be provided, then can we make $Omega$ 'nice enough' so that it does?










      share|cite|improve this question













      I have seen the Uniform Distribution/a uniform random variable for some interval in $mathbb{R}$. For example $U(a,b)$ has probability density function $frac{1}{b-a}$ (noting this is the 'volume' of the interval)



      My question : Is there a general Uniform Distribution/a general uniform random variable for a set $Omega$ which is a subset of $mathbb{R}^{n}$ , I assume if so the p.d.f will be the volume of $Omega$ ?



      If this exists could anyone tell me some of its properties like its mean, its variance, its p.d.f, /any other interesting things about it?



      If this distribution (or random variable) has not been extensively studied and the above properties can not be provided, then can we make $Omega$ 'nice enough' so that it does?







      probability-distributions uniform-distribution density-function






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      asked Nov 17 at 16:16









      Monty

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          Its PDF would be $1/mathrm{Vol}Omega$ for $x in Omega$ and $0$ otherwise.



          Its mean, variance, and other properties would all depend on your choice of $Omega$. Without a fixed $Omega$, you couldn't say anything interesting about mean, variance, etc.



          This is because the PDF is still defined on all of $R^n$ (it's just $0$). It's not really uniform in that $Omega$ can be highly irregular, not connected, etc. So what you're asking is "given an arbitrary set $Omega subset R^n$, find its center of mass, variance, and other properties". It's pretty easy to see that you can't say much without a specific set.






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            up vote
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            Its PDF would be $1/mathrm{Vol}Omega$ for $x in Omega$ and $0$ otherwise.



            Its mean, variance, and other properties would all depend on your choice of $Omega$. Without a fixed $Omega$, you couldn't say anything interesting about mean, variance, etc.



            This is because the PDF is still defined on all of $R^n$ (it's just $0$). It's not really uniform in that $Omega$ can be highly irregular, not connected, etc. So what you're asking is "given an arbitrary set $Omega subset R^n$, find its center of mass, variance, and other properties". It's pretty easy to see that you can't say much without a specific set.






            share|cite|improve this answer

























              up vote
              1
              down vote













              Its PDF would be $1/mathrm{Vol}Omega$ for $x in Omega$ and $0$ otherwise.



              Its mean, variance, and other properties would all depend on your choice of $Omega$. Without a fixed $Omega$, you couldn't say anything interesting about mean, variance, etc.



              This is because the PDF is still defined on all of $R^n$ (it's just $0$). It's not really uniform in that $Omega$ can be highly irregular, not connected, etc. So what you're asking is "given an arbitrary set $Omega subset R^n$, find its center of mass, variance, and other properties". It's pretty easy to see that you can't say much without a specific set.






              share|cite|improve this answer























                up vote
                1
                down vote










                up vote
                1
                down vote









                Its PDF would be $1/mathrm{Vol}Omega$ for $x in Omega$ and $0$ otherwise.



                Its mean, variance, and other properties would all depend on your choice of $Omega$. Without a fixed $Omega$, you couldn't say anything interesting about mean, variance, etc.



                This is because the PDF is still defined on all of $R^n$ (it's just $0$). It's not really uniform in that $Omega$ can be highly irregular, not connected, etc. So what you're asking is "given an arbitrary set $Omega subset R^n$, find its center of mass, variance, and other properties". It's pretty easy to see that you can't say much without a specific set.






                share|cite|improve this answer












                Its PDF would be $1/mathrm{Vol}Omega$ for $x in Omega$ and $0$ otherwise.



                Its mean, variance, and other properties would all depend on your choice of $Omega$. Without a fixed $Omega$, you couldn't say anything interesting about mean, variance, etc.



                This is because the PDF is still defined on all of $R^n$ (it's just $0$). It's not really uniform in that $Omega$ can be highly irregular, not connected, etc. So what you're asking is "given an arbitrary set $Omega subset R^n$, find its center of mass, variance, and other properties". It's pretty easy to see that you can't say much without a specific set.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 17 at 16:28









                helper

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                524212






























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