If $X$ is integrable then $mathbb{E}[sqrt{a_0 + a_1X^2}]$ is finite
1
$begingroup$
Let $X$ be an integrable random variable with possibly no moment of order 2.
Let $a_0$ and $a_1$ be two positive scalars. I want to prove that $mathbb{E}[sqrt{a_0 + a_1X^2}] < infty$
I tried to use Jensen or $sqrt{x} < 1 + x$, but to do that I have to assume that $mathbb{E}[X^2] < infty$ which is not the case.
probability-theory inequality
edited Dec 13 '18 at 16:06
Did
249k23227466
asked Dec 13 '18 at 15:45
Dimitri MeunierDimitri Meunier
1219
$endgroup$
add a comment |
1
$begingroup$
Let $X$ be an integrable random variable with possibly no moment of order 2.
Let $a_0$ and $a_1$ be two positive scalars. I want to prove that $mathbb{E}[sqrt{a_0 + a_1X^2}] < infty$
I tried to use Jensen or $sqrt{x} < 1 + x$, but to do that I have to assume that $mathbb{E}[X^2] < infty$ which is not the case.
probability-theory inequality
edited Dec 13 '18 at 16:06
Did
249k23227466
asked Dec 13 '18 at 15:45
Dimitri MeunierDimitri Meunier
1219
$endgroup$
2
$begingroup$
Hint: Prove the pointwise inequality $$sqrt{a_0+a_1X^2}leqslantsqrt{a_0}+sqrt{a_1},|X|$$ and integrate it.
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
7 minutes. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
Thanks, I proved $sqrt{a+x} leq sqrt{a} + sqrt{x}$ with $a geq 0$ and $x geq 0$ and it works
$endgroup$
– Dimitri Meunier
Dec 13 '18 at 16:16
$begingroup$
Exactly. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:18
add a comment |
1
1
1
$begingroup$
Let $X$ be an integrable random variable with possibly no moment of order 2.
Let $a_0$ and $a_1$ be two positive scalars. I want to prove that $mathbb{E}[sqrt{a_0 + a_1X^2}] < infty$
I tried to use Jensen or $sqrt{x} < 1 + x$, but to do that I have to assume that $mathbb{E}[X^2] < infty$ which is not the case.
probability-theory inequality
edited Dec 13 '18 at 16:06
Did
249k23227466
asked Dec 13 '18 at 15:45
Dimitri MeunierDimitri Meunier
1219
$endgroup$
Let $X$ be an integrable random variable with possibly no moment of order 2.
Let $a_0$ and $a_1$ be two positive scalars. I want to prove that $mathbb{E}[sqrt{a_0 + a_1X^2}] < infty$
I tried to use Jensen or $sqrt{x} < 1 + x$, but to do that I have to assume that $mathbb{E}[X^2] < infty$ which is not the case.
probability-theory inequality
probability-theory inequality
edited Dec 13 '18 at 16:06
Did
249k23227466
asked Dec 13 '18 at 15:45
Dimitri MeunierDimitri Meunier
1219
edited Dec 13 '18 at 16:06
Did
249k23227466
asked Dec 13 '18 at 15:45
Dimitri MeunierDimitri Meunier
1219
edited Dec 13 '18 at 16:06
Did
249k23227466
edited Dec 13 '18 at 16:06
Did
249k23227466
edited Dec 13 '18 at 16:06
Did
249k23227466
249k23227466
asked Dec 13 '18 at 15:45
Dimitri MeunierDimitri Meunier
1219
asked Dec 13 '18 at 15:45
Dimitri MeunierDimitri Meunier
1219
asked Dec 13 '18 at 15:45
Dimitri MeunierDimitri Meunier
1219
1219
2
$begingroup$
Hint: Prove the pointwise inequality $$sqrt{a_0+a_1X^2}leqslantsqrt{a_0}+sqrt{a_1},|X|$$ and integrate it.
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
7 minutes. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
Thanks, I proved $sqrt{a+x} leq sqrt{a} + sqrt{x}$ with $a geq 0$ and $x geq 0$ and it works
$endgroup$
– Dimitri Meunier
Dec 13 '18 at 16:16
$begingroup$
Exactly. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:18
add a comment |
2
$begingroup$
Hint: Prove the pointwise inequality $$sqrt{a_0+a_1X^2}leqslantsqrt{a_0}+sqrt{a_1},|X|$$ and integrate it.
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
7 minutes. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
Thanks, I proved $sqrt{a+x} leq sqrt{a} + sqrt{x}$ with $a geq 0$ and $x geq 0$ and it works
$endgroup$
– Dimitri Meunier
Dec 13 '18 at 16:16
$begingroup$
Exactly. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:18
2
2
$begingroup$
Hint: Prove the pointwise inequality $$sqrt{a_0+a_1X^2}leqslantsqrt{a_0}+sqrt{a_1},|X|$$ and integrate it.
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
Hint: Prove the pointwise inequality $$sqrt{a_0+a_1X^2}leqslantsqrt{a_0}+sqrt{a_1},|X|$$ and integrate it.
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
7 minutes. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
7 minutes. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
Thanks, I proved $sqrt{a+x} leq sqrt{a} + sqrt{x}$ with $a geq 0$ and $x geq 0$ and it works
$endgroup$
– Dimitri Meunier
Dec 13 '18 at 16:16
$begingroup$
Thanks, I proved $sqrt{a+x} leq sqrt{a} + sqrt{x}$ with $a geq 0$ and $x geq 0$ and it works
$endgroup$
– Dimitri Meunier
Dec 13 '18 at 16:16
$begingroup$
Exactly. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:18
$begingroup$
Exactly. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:18
add a comment |
1 Answer
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active
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$begingroup$
Note that $sqrt{a_1}|X|lesqrt{a_0+a_1X^2}$ implies $E|X|<infty$ is a necessary condition. That it is also sufficient follows from $0lesqrt{a_0+a_1X^2}lesqrt{a_0}+sqrt{a_1}|X|$.
answered Dec 13 '18 at 15:53
J.G.J.G.
32.8k23250
$endgroup$
$begingroup$
Interesting mistake: the fact that $Uleqslantmax{V,W}$ almost surely does not imply that $E(U)leqslantmax{E(V),E(W)}$. Thus, this step in your answer is wrong.
$endgroup$
– Did
Dec 13 '18 at 16:02
$begingroup$
And I fail to understand what you mean by: "...$E|X|$ is finite, which is necessary in the case $a_0=a_1$."
$endgroup$
– Did
Dec 13 '18 at 16:03
$begingroup$
And to be honest, I also fail to understand the two instant upvotes.
$endgroup$
– Did
Dec 13 '18 at 16:06
$begingroup$
"There's nothing wrong about these manipulations of these non-negative quantities" ?? Exercise: Find some almost surely nonnegative random variables $U$, $V$ and $W$ such that $Uleqslantmax{V,W}$ and $E(U)>max{E(V),E(W)}$.
$endgroup$
– Did
Dec 13 '18 at 16:16
$begingroup$
"final comment explains why a finite E|X| is also necessary" Except that nobody assumes $a_0=a_1$ here, so, what are you saying?
$endgroup$
– Did
Dec 13 '18 at 16:17
|
show 4 more comments
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1 Answer
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1
$begingroup$
Note that $sqrt{a_1}|X|lesqrt{a_0+a_1X^2}$ implies $E|X|<infty$ is a necessary condition. That it is also sufficient follows from $0lesqrt{a_0+a_1X^2}lesqrt{a_0}+sqrt{a_1}|X|$.
answered Dec 13 '18 at 15:53
J.G.J.G.
32.8k23250
$endgroup$
$begingroup$
Interesting mistake: the fact that $Uleqslantmax{V,W}$ almost surely does not imply that $E(U)leqslantmax{E(V),E(W)}$. Thus, this step in your answer is wrong.
$endgroup$
– Did
Dec 13 '18 at 16:02
$begingroup$
And I fail to understand what you mean by: "...$E|X|$ is finite, which is necessary in the case $a_0=a_1$."
$endgroup$
– Did
Dec 13 '18 at 16:03
$begingroup$
And to be honest, I also fail to understand the two instant upvotes.
$endgroup$
– Did
Dec 13 '18 at 16:06
$begingroup$
"There's nothing wrong about these manipulations of these non-negative quantities" ?? Exercise: Find some almost surely nonnegative random variables $U$, $V$ and $W$ such that $Uleqslantmax{V,W}$ and $E(U)>max{E(V),E(W)}$.
$endgroup$
– Did
Dec 13 '18 at 16:16
$begingroup$
"final comment explains why a finite E|X| is also necessary" Except that nobody assumes $a_0=a_1$ here, so, what are you saying?
$endgroup$
– Did
Dec 13 '18 at 16:17
|
show 4 more comments
1
$begingroup$
Note that $sqrt{a_1}|X|lesqrt{a_0+a_1X^2}$ implies $E|X|<infty$ is a necessary condition. That it is also sufficient follows from $0lesqrt{a_0+a_1X^2}lesqrt{a_0}+sqrt{a_1}|X|$.
answered Dec 13 '18 at 15:53
J.G.J.G.
32.8k23250
$endgroup$
$begingroup$
Interesting mistake: the fact that $Uleqslantmax{V,W}$ almost surely does not imply that $E(U)leqslantmax{E(V),E(W)}$. Thus, this step in your answer is wrong.
$endgroup$
– Did
Dec 13 '18 at 16:02
$begingroup$
And I fail to understand what you mean by: "...$E|X|$ is finite, which is necessary in the case $a_0=a_1$."
$endgroup$
– Did
Dec 13 '18 at 16:03
$begingroup$
And to be honest, I also fail to understand the two instant upvotes.
$endgroup$
– Did
Dec 13 '18 at 16:06
$begingroup$
"There's nothing wrong about these manipulations of these non-negative quantities" ?? Exercise: Find some almost surely nonnegative random variables $U$, $V$ and $W$ such that $Uleqslantmax{V,W}$ and $E(U)>max{E(V),E(W)}$.
$endgroup$
– Did
Dec 13 '18 at 16:16
$begingroup$
"final comment explains why a finite E|X| is also necessary" Except that nobody assumes $a_0=a_1$ here, so, what are you saying?
$endgroup$
– Did
Dec 13 '18 at 16:17
|
show 4 more comments
1
1
1
$begingroup$
Note that $sqrt{a_1}|X|lesqrt{a_0+a_1X^2}$ implies $E|X|<infty$ is a necessary condition. That it is also sufficient follows from $0lesqrt{a_0+a_1X^2}lesqrt{a_0}+sqrt{a_1}|X|$.
answered Dec 13 '18 at 15:53
J.G.J.G.
32.8k23250
$endgroup$
Note that $sqrt{a_1}|X|lesqrt{a_0+a_1X^2}$ implies $E|X|<infty$ is a necessary condition. That it is also sufficient follows from $0lesqrt{a_0+a_1X^2}lesqrt{a_0}+sqrt{a_1}|X|$.
answered Dec 13 '18 at 15:53
J.G.J.G.
32.8k23250
edited Dec 13 '18 at 16:42
answered Dec 13 '18 at 15:53
J.G.J.G.
32.8k23250
answered Dec 13 '18 at 15:53
J.G.J.G.
32.8k23250
answered Dec 13 '18 at 15:53
J.G.J.G.
32.8k23250
32.8k23250
$begingroup$
Interesting mistake: the fact that $Uleqslantmax{V,W}$ almost surely does not imply that $E(U)leqslantmax{E(V),E(W)}$. Thus, this step in your answer is wrong.
$endgroup$
– Did
Dec 13 '18 at 16:02
$begingroup$
And I fail to understand what you mean by: "...$E|X|$ is finite, which is necessary in the case $a_0=a_1$."
$endgroup$
– Did
Dec 13 '18 at 16:03
$begingroup$
And to be honest, I also fail to understand the two instant upvotes.
$endgroup$
– Did
Dec 13 '18 at 16:06
$begingroup$
"There's nothing wrong about these manipulations of these non-negative quantities" ?? Exercise: Find some almost surely nonnegative random variables $U$, $V$ and $W$ such that $Uleqslantmax{V,W}$ and $E(U)>max{E(V),E(W)}$.
$endgroup$
– Did
Dec 13 '18 at 16:16
$begingroup$
"final comment explains why a finite E|X| is also necessary" Except that nobody assumes $a_0=a_1$ here, so, what are you saying?
$endgroup$
– Did
Dec 13 '18 at 16:17
|
show 4 more comments
$begingroup$
Interesting mistake: the fact that $Uleqslantmax{V,W}$ almost surely does not imply that $E(U)leqslantmax{E(V),E(W)}$. Thus, this step in your answer is wrong.
$endgroup$
– Did
Dec 13 '18 at 16:02
$begingroup$
And I fail to understand what you mean by: "...$E|X|$ is finite, which is necessary in the case $a_0=a_1$."
$endgroup$
– Did
Dec 13 '18 at 16:03
$begingroup$
And to be honest, I also fail to understand the two instant upvotes.
$endgroup$
– Did
Dec 13 '18 at 16:06
$begingroup$
"There's nothing wrong about these manipulations of these non-negative quantities" ?? Exercise: Find some almost surely nonnegative random variables $U$, $V$ and $W$ such that $Uleqslantmax{V,W}$ and $E(U)>max{E(V),E(W)}$.
$endgroup$
– Did
Dec 13 '18 at 16:16
$begingroup$
"final comment explains why a finite E|X| is also necessary" Except that nobody assumes $a_0=a_1$ here, so, what are you saying?
$endgroup$
– Did
Dec 13 '18 at 16:17
$begingroup$
Interesting mistake: the fact that $Uleqslantmax{V,W}$ almost surely does not imply that $E(U)leqslantmax{E(V),E(W)}$. Thus, this step in your answer is wrong.
$endgroup$
– Did
Dec 13 '18 at 16:02
$begingroup$
Interesting mistake: the fact that $Uleqslantmax{V,W}$ almost surely does not imply that $E(U)leqslantmax{E(V),E(W)}$. Thus, this step in your answer is wrong.
$endgroup$
– Did
Dec 13 '18 at 16:02
$begingroup$
And I fail to understand what you mean by: "...$E|X|$ is finite, which is necessary in the case $a_0=a_1$."
$endgroup$
– Did
Dec 13 '18 at 16:03
$begingroup$
And I fail to understand what you mean by: "...$E|X|$ is finite, which is necessary in the case $a_0=a_1$."
$endgroup$
– Did
Dec 13 '18 at 16:03
$begingroup$
And to be honest, I also fail to understand the two instant upvotes.
$endgroup$
– Did
Dec 13 '18 at 16:06
$begingroup$
And to be honest, I also fail to understand the two instant upvotes.
$endgroup$
– Did
Dec 13 '18 at 16:06
$begingroup$
"There's nothing wrong about these manipulations of these non-negative quantities" ?? Exercise: Find some almost surely nonnegative random variables $U$, $V$ and $W$ such that $Uleqslantmax{V,W}$ and $E(U)>max{E(V),E(W)}$.
$endgroup$
– Did
Dec 13 '18 at 16:16
$begingroup$
"There's nothing wrong about these manipulations of these non-negative quantities" ?? Exercise: Find some almost surely nonnegative random variables $U$, $V$ and $W$ such that $Uleqslantmax{V,W}$ and $E(U)>max{E(V),E(W)}$.
$endgroup$
– Did
Dec 13 '18 at 16:16
$begingroup$
"final comment explains why a finite E|X| is also necessary" Except that nobody assumes $a_0=a_1$ here, so, what are you saying?
$endgroup$
– Did
Dec 13 '18 at 16:17
$begingroup$
"final comment explains why a finite E|X| is also necessary" Except that nobody assumes $a_0=a_1$ here, so, what are you saying?
$endgroup$
– Did
Dec 13 '18 at 16:17
|
show 4 more comments
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$begingroup$
Hint: Prove the pointwise inequality $$sqrt{a_0+a_1X^2}leqslantsqrt{a_0}+sqrt{a_1},|X|$$ and integrate it.
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
7 minutes. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:05
$begingroup$
Thanks, I proved $sqrt{a+x} leq sqrt{a} + sqrt{x}$ with $a geq 0$ and $x geq 0$ and it works
$endgroup$
– Dimitri Meunier
Dec 13 '18 at 16:16
$begingroup$
Exactly. $ $ $ $
$endgroup$
– Did
Dec 13 '18 at 16:18