Question arising from the proof of the existence of $mathbb E[X]$ when $X$ ~ $mathcal{N}(mu, sigma ^2)$
$begingroup$
$mathbb E[X]$ when $X$ ~ $mathcal{N}(mu, sigma ^2)$
I know that $mathbb E[X]$ exists iff $mathbb E[|X|]<infty$
$mathbb E[|X|]=int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx$
My Professor then goes into the next step saying:
$int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dxleqint_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx+mu$
and this is where I get confused.
I recognize that he is trying to get to a substitution $y=x-mu$, and $mathbb E[|X|]=mathbb E[|X-mu+mu|]leqmathbb E[|X-mu|]+mu$
But why is $mathbb E[|X-mu|]=int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx$ and not equal $int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-2mu)}{2sigma^2}}dx$. I mean surely the density function $frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-2mu)}{2sigma^2}}$ shifts to the left if $mu$ is subtracted.
Additional Question out of interest:
If I can prove $mathbb E[X] in mathbb R$, can I then automatically assume $mathbb E[X]$ exists? In other words, why is $mathbb E[|X|]<infty$ neccessary rather than $mathbb E[X]$? I realize that it has something to do with the Lebesgue Integral that only looks at positive functions, but I am not sure.
probability normal-distribution expected-value
$endgroup$
add a comment |
$begingroup$
$mathbb E[X]$ when $X$ ~ $mathcal{N}(mu, sigma ^2)$
I know that $mathbb E[X]$ exists iff $mathbb E[|X|]<infty$
$mathbb E[|X|]=int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx$
My Professor then goes into the next step saying:
$int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dxleqint_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx+mu$
and this is where I get confused.
I recognize that he is trying to get to a substitution $y=x-mu$, and $mathbb E[|X|]=mathbb E[|X-mu+mu|]leqmathbb E[|X-mu|]+mu$
But why is $mathbb E[|X-mu|]=int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx$ and not equal $int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-2mu)}{2sigma^2}}dx$. I mean surely the density function $frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-2mu)}{2sigma^2}}$ shifts to the left if $mu$ is subtracted.
Additional Question out of interest:
If I can prove $mathbb E[X] in mathbb R$, can I then automatically assume $mathbb E[X]$ exists? In other words, why is $mathbb E[|X|]<infty$ neccessary rather than $mathbb E[X]$? I realize that it has something to do with the Lebesgue Integral that only looks at positive functions, but I am not sure.
probability normal-distribution expected-value
$endgroup$
add a comment |
$begingroup$
$mathbb E[X]$ when $X$ ~ $mathcal{N}(mu, sigma ^2)$
I know that $mathbb E[X]$ exists iff $mathbb E[|X|]<infty$
$mathbb E[|X|]=int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx$
My Professor then goes into the next step saying:
$int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dxleqint_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx+mu$
and this is where I get confused.
I recognize that he is trying to get to a substitution $y=x-mu$, and $mathbb E[|X|]=mathbb E[|X-mu+mu|]leqmathbb E[|X-mu|]+mu$
But why is $mathbb E[|X-mu|]=int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx$ and not equal $int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-2mu)}{2sigma^2}}dx$. I mean surely the density function $frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-2mu)}{2sigma^2}}$ shifts to the left if $mu$ is subtracted.
Additional Question out of interest:
If I can prove $mathbb E[X] in mathbb R$, can I then automatically assume $mathbb E[X]$ exists? In other words, why is $mathbb E[|X|]<infty$ neccessary rather than $mathbb E[X]$? I realize that it has something to do with the Lebesgue Integral that only looks at positive functions, but I am not sure.
probability normal-distribution expected-value
$endgroup$
$mathbb E[X]$ when $X$ ~ $mathcal{N}(mu, sigma ^2)$
I know that $mathbb E[X]$ exists iff $mathbb E[|X|]<infty$
$mathbb E[|X|]=int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx$
My Professor then goes into the next step saying:
$int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dxleqint_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx+mu$
and this is where I get confused.
I recognize that he is trying to get to a substitution $y=x-mu$, and $mathbb E[|X|]=mathbb E[|X-mu+mu|]leqmathbb E[|X-mu|]+mu$
But why is $mathbb E[|X-mu|]=int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx$ and not equal $int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-2mu)}{2sigma^2}}dx$. I mean surely the density function $frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-2mu)}{2sigma^2}}$ shifts to the left if $mu$ is subtracted.
Additional Question out of interest:
If I can prove $mathbb E[X] in mathbb R$, can I then automatically assume $mathbb E[X]$ exists? In other words, why is $mathbb E[|X|]<infty$ neccessary rather than $mathbb E[X]$? I realize that it has something to do with the Lebesgue Integral that only looks at positive functions, but I am not sure.
probability normal-distribution expected-value
probability normal-distribution expected-value
asked Dec 14 '18 at 16:46
SABOYSABOY
694311
694311
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$begingroup$
What your professor did is simply applying the triangle inequality (assuming $mugeq 0$)
$$
|x|leq|x-mu|+mu,
$$
which implies that
begin{align}
int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
&leq
int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
+int_{mathbb R}mufrac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx\
&=
int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
+mu.
end{align}
For the additional question: lots of authors define $EX$ only for absolutely integrable random variables (i.e., $E|X|<infty$) and leave $EX$ undefined when $E|X|=infty$. Whenever $EX$ is defined, it must be a real number (in the context of real random variables).
$endgroup$
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
What your professor did is simply applying the triangle inequality (assuming $mugeq 0$)
$$
|x|leq|x-mu|+mu,
$$
which implies that
begin{align}
int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
&leq
int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
+int_{mathbb R}mufrac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx\
&=
int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
+mu.
end{align}
For the additional question: lots of authors define $EX$ only for absolutely integrable random variables (i.e., $E|X|<infty$) and leave $EX$ undefined when $E|X|=infty$. Whenever $EX$ is defined, it must be a real number (in the context of real random variables).
$endgroup$
add a comment |
$begingroup$
What your professor did is simply applying the triangle inequality (assuming $mugeq 0$)
$$
|x|leq|x-mu|+mu,
$$
which implies that
begin{align}
int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
&leq
int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
+int_{mathbb R}mufrac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx\
&=
int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
+mu.
end{align}
For the additional question: lots of authors define $EX$ only for absolutely integrable random variables (i.e., $E|X|<infty$) and leave $EX$ undefined when $E|X|=infty$. Whenever $EX$ is defined, it must be a real number (in the context of real random variables).
$endgroup$
add a comment |
$begingroup$
What your professor did is simply applying the triangle inequality (assuming $mugeq 0$)
$$
|x|leq|x-mu|+mu,
$$
which implies that
begin{align}
int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
&leq
int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
+int_{mathbb R}mufrac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx\
&=
int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
+mu.
end{align}
For the additional question: lots of authors define $EX$ only for absolutely integrable random variables (i.e., $E|X|<infty$) and leave $EX$ undefined when $E|X|=infty$. Whenever $EX$ is defined, it must be a real number (in the context of real random variables).
$endgroup$
What your professor did is simply applying the triangle inequality (assuming $mugeq 0$)
$$
|x|leq|x-mu|+mu,
$$
which implies that
begin{align}
int_{mathbb R}|x|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
&leq
int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
+int_{mathbb R}mufrac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx\
&=
int_{mathbb R}|x-mu|frac{1}{sqrt{2pi sigma^2}}e^{-frac{(x-mu)}{2sigma^2}}dx
+mu.
end{align}
For the additional question: lots of authors define $EX$ only for absolutely integrable random variables (i.e., $E|X|<infty$) and leave $EX$ undefined when $E|X|=infty$. Whenever $EX$ is defined, it must be a real number (in the context of real random variables).
edited Dec 14 '18 at 17:14
answered Dec 14 '18 at 17:05
user587192
add a comment |
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