How to solve expectation of continuous variables
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If it is assumed that $xthicksim N(0,sigma^2)$, then it can be shown that $E{exp(x)}=exp(0.5sigma^2)$
How to show that?
probability
asked Nov 29 '18 at 22:06
Yao ZhaoYao Zhao
215
$endgroup$
add a comment |
$begingroup$
If it is assumed that $xthicksim N(0,sigma^2)$, then it can be shown that $E{exp(x)}=exp(0.5sigma^2)$
How to show that?
probability
asked Nov 29 '18 at 22:06
Yao ZhaoYao Zhao
215
$endgroup$
$begingroup$
If $X$ has a normal distribution then $exp(X)$ has a log-normal distribution
$endgroup$
– Henry
Nov 29 '18 at 22:11
$begingroup$
what's the meaning of log-normal distribution?
$endgroup$
– Yao Zhao
Nov 29 '18 at 22:12
$begingroup$
en.wikipedia.org/wiki/Log-normal_distribution
$endgroup$
– Henry
Nov 29 '18 at 22:12
add a comment |
$begingroup$
If it is assumed that $xthicksim N(0,sigma^2)$, then it can be shown that $E{exp(x)}=exp(0.5sigma^2)$
How to show that?
probability
asked Nov 29 '18 at 22:06
Yao ZhaoYao Zhao
215
$endgroup$
If it is assumed that $xthicksim N(0,sigma^2)$, then it can be shown that $E{exp(x)}=exp(0.5sigma^2)$
How to show that?
probability
probability
asked Nov 29 '18 at 22:06
Yao ZhaoYao Zhao
215
asked Nov 29 '18 at 22:06
Yao ZhaoYao Zhao
215
asked Nov 29 '18 at 22:06
Yao ZhaoYao Zhao
215
asked Nov 29 '18 at 22:06
Yao ZhaoYao Zhao
215
asked Nov 29 '18 at 22:06
Yao ZhaoYao Zhao
215
215
$begingroup$
If $X$ has a normal distribution then $exp(X)$ has a log-normal distribution
$endgroup$
– Henry
Nov 29 '18 at 22:11
$begingroup$
what's the meaning of log-normal distribution?
$endgroup$
– Yao Zhao
Nov 29 '18 at 22:12
$begingroup$
en.wikipedia.org/wiki/Log-normal_distribution
$endgroup$
– Henry
Nov 29 '18 at 22:12
add a comment |
$begingroup$
If $X$ has a normal distribution then $exp(X)$ has a log-normal distribution
$endgroup$
– Henry
Nov 29 '18 at 22:11
$begingroup$
what's the meaning of log-normal distribution?
$endgroup$
– Yao Zhao
Nov 29 '18 at 22:12
$begingroup$
en.wikipedia.org/wiki/Log-normal_distribution
$endgroup$
– Henry
Nov 29 '18 at 22:12
$begingroup$
If $X$ has a normal distribution then $exp(X)$ has a log-normal distribution
$endgroup$
– Henry
Nov 29 '18 at 22:11
$begingroup$
If $X$ has a normal distribution then $exp(X)$ has a log-normal distribution
$endgroup$
– Henry
Nov 29 '18 at 22:11
$begingroup$
what's the meaning of log-normal distribution?
$endgroup$
– Yao Zhao
Nov 29 '18 at 22:12
$begingroup$
what's the meaning of log-normal distribution?
$endgroup$
– Yao Zhao
Nov 29 '18 at 22:12
$begingroup$
en.wikipedia.org/wiki/Log-normal_distribution
$endgroup$
– Henry
Nov 29 '18 at 22:12
$begingroup$
en.wikipedia.org/wiki/Log-normal_distribution
$endgroup$
– Henry
Nov 29 '18 at 22:12
add a comment |
1 Answer
1
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votes
$begingroup$
$Eexp tX$ is called the moment-generating function or mgf of $X$. We can prove $Eexp tX=exp sigma^2t^2/2$ for this distribution. The expectation is this integral (note the substitution $y=x-sigma^2 t$): $$int_{Bbb R}frac{1}{sigmasqrt{2pi}}expbigg(tx-frac{x^2}{2sigma^2}bigg)dx=int_{Bbb R}frac{1}{sigmasqrt{2pi}}expbigg(frac{sigma^4t^2-y^2}{2sigma^2}bigg)dy=exp sigma^2 t^2/2.$$
answered Nov 29 '18 at 22:22
J.G.J.G.
24.1k22539
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
$Eexp tX$ is called the moment-generating function or mgf of $X$. We can prove $Eexp tX=exp sigma^2t^2/2$ for this distribution. The expectation is this integral (note the substitution $y=x-sigma^2 t$): $$int_{Bbb R}frac{1}{sigmasqrt{2pi}}expbigg(tx-frac{x^2}{2sigma^2}bigg)dx=int_{Bbb R}frac{1}{sigmasqrt{2pi}}expbigg(frac{sigma^4t^2-y^2}{2sigma^2}bigg)dy=exp sigma^2 t^2/2.$$
answered Nov 29 '18 at 22:22
J.G.J.G.
24.1k22539
$endgroup$
add a comment |
$begingroup$
$Eexp tX$ is called the moment-generating function or mgf of $X$. We can prove $Eexp tX=exp sigma^2t^2/2$ for this distribution. The expectation is this integral (note the substitution $y=x-sigma^2 t$): $$int_{Bbb R}frac{1}{sigmasqrt{2pi}}expbigg(tx-frac{x^2}{2sigma^2}bigg)dx=int_{Bbb R}frac{1}{sigmasqrt{2pi}}expbigg(frac{sigma^4t^2-y^2}{2sigma^2}bigg)dy=exp sigma^2 t^2/2.$$
answered Nov 29 '18 at 22:22
J.G.J.G.
24.1k22539
$endgroup$
add a comment |
$begingroup$
$Eexp tX$ is called the moment-generating function or mgf of $X$. We can prove $Eexp tX=exp sigma^2t^2/2$ for this distribution. The expectation is this integral (note the substitution $y=x-sigma^2 t$): $$int_{Bbb R}frac{1}{sigmasqrt{2pi}}expbigg(tx-frac{x^2}{2sigma^2}bigg)dx=int_{Bbb R}frac{1}{sigmasqrt{2pi}}expbigg(frac{sigma^4t^2-y^2}{2sigma^2}bigg)dy=exp sigma^2 t^2/2.$$
answered Nov 29 '18 at 22:22
J.G.J.G.
24.1k22539
$endgroup$
$Eexp tX$ is called the moment-generating function or mgf of $X$. We can prove $Eexp tX=exp sigma^2t^2/2$ for this distribution. The expectation is this integral (note the substitution $y=x-sigma^2 t$): $$int_{Bbb R}frac{1}{sigmasqrt{2pi}}expbigg(tx-frac{x^2}{2sigma^2}bigg)dx=int_{Bbb R}frac{1}{sigmasqrt{2pi}}expbigg(frac{sigma^4t^2-y^2}{2sigma^2}bigg)dy=exp sigma^2 t^2/2.$$
answered Nov 29 '18 at 22:22
J.G.J.G.
24.1k22539
answered Nov 29 '18 at 22:22
J.G.J.G.
24.1k22539
answered Nov 29 '18 at 22:22
J.G.J.G.
24.1k22539
answered Nov 29 '18 at 22:22
J.G.J.G.
24.1k22539
24.1k22539
add a comment |
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$begingroup$
If $X$ has a normal distribution then $exp(X)$ has a log-normal distribution
$endgroup$
– Henry
Nov 29 '18 at 22:11
$begingroup$
what's the meaning of log-normal distribution?
$endgroup$
– Yao Zhao
Nov 29 '18 at 22:12
$begingroup$
en.wikipedia.org/wiki/Log-normal_distribution
$endgroup$
– Henry
Nov 29 '18 at 22:12