Mixing
explanation of E[X] = Sup(E[Y] : Y a simple r.v.)
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Could someone explain me the meaning of the following expected value of a positive random variables $X$?
$mathbb E[X] = sup({mathbb E[Y] : Ytext{ a simple r.v. with }0 < Y < X})$
where simple random variable means:
$$Y=sum_{i=1}^n a_imathbf1_{A_i}$$
supremum-and-infimum expected-value
edited Jan 14 at 11:49
drhab
101k545136
asked Jan 14 at 11:33
AlbertAlbert
11
$endgroup$
add a comment |
$begingroup$
Could someone explain me the meaning of the following expected value of a positive random variables $X$?
$mathbb E[X] = sup({mathbb E[Y] : Ytext{ a simple r.v. with }0 < Y < X})$
where simple random variable means:
$$Y=sum_{i=1}^n a_imathbf1_{A_i}$$
supremum-and-infimum expected-value
edited Jan 14 at 11:49
drhab
101k545136
asked Jan 14 at 11:33
AlbertAlbert
11
$endgroup$
add a comment |
$begingroup$
Could someone explain me the meaning of the following expected value of a positive random variables $X$?
$mathbb E[X] = sup({mathbb E[Y] : Ytext{ a simple r.v. with }0 < Y < X})$
where simple random variable means:
$$Y=sum_{i=1}^n a_imathbf1_{A_i}$$
supremum-and-infimum expected-value
edited Jan 14 at 11:49
drhab
101k545136
asked Jan 14 at 11:33
AlbertAlbert
11
$endgroup$
Could someone explain me the meaning of the following expected value of a positive random variables $X$?
$mathbb E[X] = sup({mathbb E[Y] : Ytext{ a simple r.v. with }0 < Y < X})$
where simple random variable means:
$$Y=sum_{i=1}^n a_imathbf1_{A_i}$$
supremum-and-infimum expected-value
supremum-and-infimum expected-value
edited Jan 14 at 11:49
drhab
101k545136
asked Jan 14 at 11:33
AlbertAlbert
11
edited Jan 14 at 11:49
drhab
101k545136
asked Jan 14 at 11:33
AlbertAlbert
11
edited Jan 14 at 11:49
drhab
101k545136
edited Jan 14 at 11:49
drhab
101k545136
edited Jan 14 at 11:49
drhab
101k545136
101k545136
asked Jan 14 at 11:33
AlbertAlbert
11
asked Jan 14 at 11:33
AlbertAlbert
11
asked Jan 14 at 11:33
AlbertAlbert
11
11
add a comment |
add a comment |
2 Answers
2
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Define $Y_n=frac {k-1} {2^{n}}$ when $frac {k-1} {2^{n}} leq X <frac k {2^{n}}$ for $1leq k <n2^{n}$ and $Y_n=n$ when $X >n$. Then $Y_n$ is a simple function, $0leq Y_n leq X$ and $Y_n$ increases to $X$ as $n$ increases to $infty$. By Monotone Convergence Theorem $EY_n to X$. It follows that $sup {EY:0leq Yleq X, Y text {simple}} geq EX$. The reverse inequality is obvious.
answered Jan 14 at 11:47
Kavi Rama MurthyKavi Rama Murthy
60.4k42161
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add a comment |
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For a nonnegative simple random variable $Y=sum_{i=1}^na_imathbf1_{A_i}$ we have a definition of its expectation like this:$$mathbb EY:=sum_{i=1}^na_iP(A_i)tag1$$What we want is a more general definition for nonnegative random variables and this can be achieved by stating that - if $X$ is a nonnegative random variable:$$mathbb EX:=sup({mathbb EYmid Ytext{ is a simple nonnegative random variable that satisfies }Yleq X})tag2$$
This gives a definition for a greater class of random variables.
Observe however that for nonnegative simple random variable $Y$ we now have $2$ definitions. It must be checked that $mathbb EY$ according to $(1)$ is the same as $mathbb EY$ according to $(2)$ (where $X$ is replaced by $Y$ and $Y$ by e.g. $Z$), and fortunately that is indeed the case.
It is a good exercise to this check yourself.
answered Jan 14 at 11:47
drhabdrhab
101k545136
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add a comment |
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2 Answers
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2 Answers
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$begingroup$
Define $Y_n=frac {k-1} {2^{n}}$ when $frac {k-1} {2^{n}} leq X <frac k {2^{n}}$ for $1leq k <n2^{n}$ and $Y_n=n$ when $X >n$. Then $Y_n$ is a simple function, $0leq Y_n leq X$ and $Y_n$ increases to $X$ as $n$ increases to $infty$. By Monotone Convergence Theorem $EY_n to X$. It follows that $sup {EY:0leq Yleq X, Y text {simple}} geq EX$. The reverse inequality is obvious.
answered Jan 14 at 11:47
Kavi Rama MurthyKavi Rama Murthy
60.4k42161
$endgroup$
add a comment |
$begingroup$
Define $Y_n=frac {k-1} {2^{n}}$ when $frac {k-1} {2^{n}} leq X <frac k {2^{n}}$ for $1leq k <n2^{n}$ and $Y_n=n$ when $X >n$. Then $Y_n$ is a simple function, $0leq Y_n leq X$ and $Y_n$ increases to $X$ as $n$ increases to $infty$. By Monotone Convergence Theorem $EY_n to X$. It follows that $sup {EY:0leq Yleq X, Y text {simple}} geq EX$. The reverse inequality is obvious.
answered Jan 14 at 11:47
Kavi Rama MurthyKavi Rama Murthy
60.4k42161
$endgroup$
add a comment |
$begingroup$
Define $Y_n=frac {k-1} {2^{n}}$ when $frac {k-1} {2^{n}} leq X <frac k {2^{n}}$ for $1leq k <n2^{n}$ and $Y_n=n$ when $X >n$. Then $Y_n$ is a simple function, $0leq Y_n leq X$ and $Y_n$ increases to $X$ as $n$ increases to $infty$. By Monotone Convergence Theorem $EY_n to X$. It follows that $sup {EY:0leq Yleq X, Y text {simple}} geq EX$. The reverse inequality is obvious.
answered Jan 14 at 11:47
Kavi Rama MurthyKavi Rama Murthy
60.4k42161
$endgroup$
Define $Y_n=frac {k-1} {2^{n}}$ when $frac {k-1} {2^{n}} leq X <frac k {2^{n}}$ for $1leq k <n2^{n}$ and $Y_n=n$ when $X >n$. Then $Y_n$ is a simple function, $0leq Y_n leq X$ and $Y_n$ increases to $X$ as $n$ increases to $infty$. By Monotone Convergence Theorem $EY_n to X$. It follows that $sup {EY:0leq Yleq X, Y text {simple}} geq EX$. The reverse inequality is obvious.
answered Jan 14 at 11:47
Kavi Rama MurthyKavi Rama Murthy
60.4k42161
answered Jan 14 at 11:47
Kavi Rama MurthyKavi Rama Murthy
60.4k42161
answered Jan 14 at 11:47
Kavi Rama MurthyKavi Rama Murthy
60.4k42161
answered Jan 14 at 11:47
Kavi Rama MurthyKavi Rama Murthy
60.4k42161
60.4k42161
add a comment |
add a comment |
$begingroup$
For a nonnegative simple random variable $Y=sum_{i=1}^na_imathbf1_{A_i}$ we have a definition of its expectation like this:$$mathbb EY:=sum_{i=1}^na_iP(A_i)tag1$$What we want is a more general definition for nonnegative random variables and this can be achieved by stating that - if $X$ is a nonnegative random variable:$$mathbb EX:=sup({mathbb EYmid Ytext{ is a simple nonnegative random variable that satisfies }Yleq X})tag2$$
This gives a definition for a greater class of random variables.
Observe however that for nonnegative simple random variable $Y$ we now have $2$ definitions. It must be checked that $mathbb EY$ according to $(1)$ is the same as $mathbb EY$ according to $(2)$ (where $X$ is replaced by $Y$ and $Y$ by e.g. $Z$), and fortunately that is indeed the case.
It is a good exercise to this check yourself.
answered Jan 14 at 11:47
drhabdrhab
101k545136
$endgroup$
add a comment |
$begingroup$
For a nonnegative simple random variable $Y=sum_{i=1}^na_imathbf1_{A_i}$ we have a definition of its expectation like this:$$mathbb EY:=sum_{i=1}^na_iP(A_i)tag1$$What we want is a more general definition for nonnegative random variables and this can be achieved by stating that - if $X$ is a nonnegative random variable:$$mathbb EX:=sup({mathbb EYmid Ytext{ is a simple nonnegative random variable that satisfies }Yleq X})tag2$$
This gives a definition for a greater class of random variables.
Observe however that for nonnegative simple random variable $Y$ we now have $2$ definitions. It must be checked that $mathbb EY$ according to $(1)$ is the same as $mathbb EY$ according to $(2)$ (where $X$ is replaced by $Y$ and $Y$ by e.g. $Z$), and fortunately that is indeed the case.
It is a good exercise to this check yourself.
answered Jan 14 at 11:47
drhabdrhab
101k545136
$endgroup$
add a comment |
$begingroup$
For a nonnegative simple random variable $Y=sum_{i=1}^na_imathbf1_{A_i}$ we have a definition of its expectation like this:$$mathbb EY:=sum_{i=1}^na_iP(A_i)tag1$$What we want is a more general definition for nonnegative random variables and this can be achieved by stating that - if $X$ is a nonnegative random variable:$$mathbb EX:=sup({mathbb EYmid Ytext{ is a simple nonnegative random variable that satisfies }Yleq X})tag2$$
This gives a definition for a greater class of random variables.
Observe however that for nonnegative simple random variable $Y$ we now have $2$ definitions. It must be checked that $mathbb EY$ according to $(1)$ is the same as $mathbb EY$ according to $(2)$ (where $X$ is replaced by $Y$ and $Y$ by e.g. $Z$), and fortunately that is indeed the case.
It is a good exercise to this check yourself.
answered Jan 14 at 11:47
drhabdrhab
101k545136
$endgroup$
For a nonnegative simple random variable $Y=sum_{i=1}^na_imathbf1_{A_i}$ we have a definition of its expectation like this:$$mathbb EY:=sum_{i=1}^na_iP(A_i)tag1$$What we want is a more general definition for nonnegative random variables and this can be achieved by stating that - if $X$ is a nonnegative random variable:$$mathbb EX:=sup({mathbb EYmid Ytext{ is a simple nonnegative random variable that satisfies }Yleq X})tag2$$
This gives a definition for a greater class of random variables.
Observe however that for nonnegative simple random variable $Y$ we now have $2$ definitions. It must be checked that $mathbb EY$ according to $(1)$ is the same as $mathbb EY$ according to $(2)$ (where $X$ is replaced by $Y$ and $Y$ by e.g. $Z$), and fortunately that is indeed the case.
It is a good exercise to this check yourself.
answered Jan 14 at 11:47
drhabdrhab
101k545136
edited Jan 14 at 11:52
answered Jan 14 at 11:47
drhabdrhab
101k545136
answered Jan 14 at 11:47
drhabdrhab
101k545136
answered Jan 14 at 11:47
drhabdrhab
101k545136
101k545136
add a comment |
add a comment |
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