What is the sample space of a function of a random variable?












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$begingroup$


I know that a function of a random variable is itself a random variable. So, if $Y$ is a random variable, then $X = f(Y)$ is also a random variable.



Assuming the sample space of $Y$ is ${y_1, y_2, ..., y_n}$, then is there some way to express the sample space of $X$ in terms of the sample space of $Y$?



If yes, can you then express $E[X]$ as $sum_{y_i}P(Y = y_i)f(y_i)$?






probability random-variables random expected-value







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  • $begingroup$
    First, a real-valued RV is a function from the sample space to $mathbb{R}$, that is $X:Omega to mathbb{R}$, so that for each $omega in Omega$ we have $X(omega)in mathbb{R}$. So in this sense, the outcomes $omega$ are not the values one writes in expressions like $mathbb{P}(X=x)$. To be fully precise, one means $$mathbb{P}({omega in Omega : X(omega)=x}),$$ whenever one writes the conventional notation $mathbb{P}(X=x)$. To repeat: outcomes $omega$ are not the same as the variates $x$ that the variable $X$ takes on.
    $endgroup$
    – LoveTooNap29
    Feb 2 at 23:20


















0












$begingroup$


I know that a function of a random variable is itself a random variable. So, if $Y$ is a random variable, then $X = f(Y)$ is also a random variable.



Assuming the sample space of $Y$ is ${y_1, y_2, ..., y_n}$, then is there some way to express the sample space of $X$ in terms of the sample space of $Y$?



If yes, can you then express $E[X]$ as $sum_{y_i}P(Y = y_i)f(y_i)$?






probability random-variables random expected-value







share|cite|improve this question









$endgroup$












  • $begingroup$
    First, a real-valued RV is a function from the sample space to $mathbb{R}$, that is $X:Omega to mathbb{R}$, so that for each $omega in Omega$ we have $X(omega)in mathbb{R}$. So in this sense, the outcomes $omega$ are not the values one writes in expressions like $mathbb{P}(X=x)$. To be fully precise, one means $$mathbb{P}({omega in Omega : X(omega)=x}),$$ whenever one writes the conventional notation $mathbb{P}(X=x)$. To repeat: outcomes $omega$ are not the same as the variates $x$ that the variable $X$ takes on.
    $endgroup$
    – LoveTooNap29
    Feb 2 at 23:20
















0












0








0





$begingroup$


I know that a function of a random variable is itself a random variable. So, if $Y$ is a random variable, then $X = f(Y)$ is also a random variable.



Assuming the sample space of $Y$ is ${y_1, y_2, ..., y_n}$, then is there some way to express the sample space of $X$ in terms of the sample space of $Y$?



If yes, can you then express $E[X]$ as $sum_{y_i}P(Y = y_i)f(y_i)$?






probability random-variables random expected-value







share|cite|improve this question









$endgroup$




I know that a function of a random variable is itself a random variable. So, if $Y$ is a random variable, then $X = f(Y)$ is also a random variable.



Assuming the sample space of $Y$ is ${y_1, y_2, ..., y_n}$, then is there some way to express the sample space of $X$ in terms of the sample space of $Y$?



If yes, can you then express $E[X]$ as $sum_{y_i}P(Y = y_i)f(y_i)$?






probability random-variables random expected-value




probability random-variables random expected-value






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Feb 2 at 23:13









KevinKevin

1




1












  • $begingroup$
    First, a real-valued RV is a function from the sample space to $mathbb{R}$, that is $X:Omega to mathbb{R}$, so that for each $omega in Omega$ we have $X(omega)in mathbb{R}$. So in this sense, the outcomes $omega$ are not the values one writes in expressions like $mathbb{P}(X=x)$. To be fully precise, one means $$mathbb{P}({omega in Omega : X(omega)=x}),$$ whenever one writes the conventional notation $mathbb{P}(X=x)$. To repeat: outcomes $omega$ are not the same as the variates $x$ that the variable $X$ takes on.
    $endgroup$
    – LoveTooNap29
    Feb 2 at 23:20




















  • $begingroup$
    First, a real-valued RV is a function from the sample space to $mathbb{R}$, that is $X:Omega to mathbb{R}$, so that for each $omega in Omega$ we have $X(omega)in mathbb{R}$. So in this sense, the outcomes $omega$ are not the values one writes in expressions like $mathbb{P}(X=x)$. To be fully precise, one means $$mathbb{P}({omega in Omega : X(omega)=x}),$$ whenever one writes the conventional notation $mathbb{P}(X=x)$. To repeat: outcomes $omega$ are not the same as the variates $x$ that the variable $X$ takes on.
    $endgroup$
    – LoveTooNap29
    Feb 2 at 23:20


















$begingroup$
First, a real-valued RV is a function from the sample space to $mathbb{R}$, that is $X:Omega to mathbb{R}$, so that for each $omega in Omega$ we have $X(omega)in mathbb{R}$. So in this sense, the outcomes $omega$ are not the values one writes in expressions like $mathbb{P}(X=x)$. To be fully precise, one means $$mathbb{P}({omega in Omega : X(omega)=x}),$$ whenever one writes the conventional notation $mathbb{P}(X=x)$. To repeat: outcomes $omega$ are not the same as the variates $x$ that the variable $X$ takes on.
$endgroup$
– LoveTooNap29
Feb 2 at 23:20






$begingroup$
First, a real-valued RV is a function from the sample space to $mathbb{R}$, that is $X:Omega to mathbb{R}$, so that for each $omega in Omega$ we have $X(omega)in mathbb{R}$. So in this sense, the outcomes $omega$ are not the values one writes in expressions like $mathbb{P}(X=x)$. To be fully precise, one means $$mathbb{P}({omega in Omega : X(omega)=x}),$$ whenever one writes the conventional notation $mathbb{P}(X=x)$. To repeat: outcomes $omega$ are not the same as the variates $x$ that the variable $X$ takes on.
$endgroup$
– LoveTooNap29
Feb 2 at 23:20












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