Find a measurable function $g:mathbb{R}tomathbb{R}$ s.t. $mathbb{E}(g(mathcal{N}(0,1)))=2$
$begingroup$
Let $Xsimmathcal{N}(0,1)$ a random variable. Find a measurable function $g:mathbb{R}tomathbb{R}$ such that $mathbb{E}(g(X))=2$.
probability random-variables normal-distribution measurable-functions
asked Jan 19 at 15:20
J. DoeJ. Doe
15210
$endgroup$
add a comment |
$begingroup$
Let $Xsimmathcal{N}(0,1)$ a random variable. Find a measurable function $g:mathbb{R}tomathbb{R}$ such that $mathbb{E}(g(X))=2$.
probability random-variables normal-distribution measurable-functions
asked Jan 19 at 15:20
J. DoeJ. Doe
15210
$endgroup$
$begingroup$
Hint: What is E(X^2)? Simpler: What is E(X)? Even simpler: What is E(1)?
$endgroup$
– Did
Jan 19 at 16:33
add a comment |
$begingroup$
Let $Xsimmathcal{N}(0,1)$ a random variable. Find a measurable function $g:mathbb{R}tomathbb{R}$ such that $mathbb{E}(g(X))=2$.
probability random-variables normal-distribution measurable-functions
asked Jan 19 at 15:20
J. DoeJ. Doe
15210
$endgroup$
Let $Xsimmathcal{N}(0,1)$ a random variable. Find a measurable function $g:mathbb{R}tomathbb{R}$ such that $mathbb{E}(g(X))=2$.
probability random-variables normal-distribution measurable-functions
probability random-variables normal-distribution measurable-functions
asked Jan 19 at 15:20
J. DoeJ. Doe
15210
asked Jan 19 at 15:20
J. DoeJ. Doe
15210
edited Jan 19 at 16:28
J. Doe
asked Jan 19 at 15:20
J. DoeJ. Doe
15210
asked Jan 19 at 15:20
J. DoeJ. Doe
15210
asked Jan 19 at 15:20
J. DoeJ. Doe
15210
15210
$begingroup$
Hint: What is E(X^2)? Simpler: What is E(X)? Even simpler: What is E(1)?
$endgroup$
– Did
Jan 19 at 16:33
add a comment |
$begingroup$
Hint: What is E(X^2)? Simpler: What is E(X)? Even simpler: What is E(1)?
$endgroup$
– Did
Jan 19 at 16:33
$begingroup$
Hint: What is E(X^2)? Simpler: What is E(X)? Even simpler: What is E(1)?
$endgroup$
– Did
Jan 19 at 16:33
$begingroup$
Hint: What is E(X^2)? Simpler: What is E(X)? Even simpler: What is E(1)?
$endgroup$
– Did
Jan 19 at 16:33
add a comment |
1 Answer
1
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votes
$begingroup$
We know that the density function of $X$ is
$f_X(s)={1oversqrt{2pi}}e^{-s^2over 2}$. Let $g:mathbb{R}tomathbb{R}$ a function defined by
$$
\ g(s)=sqrt{2pi}e^{s^2over 2}cdottextbf{1}_{{0leq sleq 2}}
$$
$smapsto textbf{1}_{{0leq s leq 2} }$ is a measurable function because it's a linear combination of indicators of borel sets, $smapstosqrt{2pi}e^{s^2over 2}$ is a measurable function because it's continuous, and $g$ is a measurable function because it's a composition of measurable functions. Thus,
$$
\ mathbb{E}(g(X))=int_{-infty}^infty g(s)f_X(s)ds=int_0^2ds=2
$$
answered Jan 19 at 15:20
J. DoeJ. Doe
15210
$endgroup$
1
$begingroup$
This is fine and dandy but far too complicated, maybe? What if $g(x) = x+2$ for all $xin mathbb R$? Is $g(x)$ measurable? What is $E[g(X)] = E[X+2]$?
$endgroup$
– Dilip Sarwate
Jan 19 at 16:22
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
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votes
$begingroup$
We know that the density function of $X$ is
$f_X(s)={1oversqrt{2pi}}e^{-s^2over 2}$. Let $g:mathbb{R}tomathbb{R}$ a function defined by
$$
\ g(s)=sqrt{2pi}e^{s^2over 2}cdottextbf{1}_{{0leq sleq 2}}
$$
$smapsto textbf{1}_{{0leq s leq 2} }$ is a measurable function because it's a linear combination of indicators of borel sets, $smapstosqrt{2pi}e^{s^2over 2}$ is a measurable function because it's continuous, and $g$ is a measurable function because it's a composition of measurable functions. Thus,
$$
\ mathbb{E}(g(X))=int_{-infty}^infty g(s)f_X(s)ds=int_0^2ds=2
$$
answered Jan 19 at 15:20
J. DoeJ. Doe
15210
$endgroup$
1
$begingroup$
This is fine and dandy but far too complicated, maybe? What if $g(x) = x+2$ for all $xin mathbb R$? Is $g(x)$ measurable? What is $E[g(X)] = E[X+2]$?
$endgroup$
– Dilip Sarwate
Jan 19 at 16:22
add a comment |
$begingroup$
We know that the density function of $X$ is
$f_X(s)={1oversqrt{2pi}}e^{-s^2over 2}$. Let $g:mathbb{R}tomathbb{R}$ a function defined by
$$
\ g(s)=sqrt{2pi}e^{s^2over 2}cdottextbf{1}_{{0leq sleq 2}}
$$
$smapsto textbf{1}_{{0leq s leq 2} }$ is a measurable function because it's a linear combination of indicators of borel sets, $smapstosqrt{2pi}e^{s^2over 2}$ is a measurable function because it's continuous, and $g$ is a measurable function because it's a composition of measurable functions. Thus,
$$
\ mathbb{E}(g(X))=int_{-infty}^infty g(s)f_X(s)ds=int_0^2ds=2
$$
answered Jan 19 at 15:20
J. DoeJ. Doe
15210
$endgroup$
1
$begingroup$
This is fine and dandy but far too complicated, maybe? What if $g(x) = x+2$ for all $xin mathbb R$? Is $g(x)$ measurable? What is $E[g(X)] = E[X+2]$?
$endgroup$
– Dilip Sarwate
Jan 19 at 16:22
add a comment |
$begingroup$
We know that the density function of $X$ is
$f_X(s)={1oversqrt{2pi}}e^{-s^2over 2}$. Let $g:mathbb{R}tomathbb{R}$ a function defined by
$$
\ g(s)=sqrt{2pi}e^{s^2over 2}cdottextbf{1}_{{0leq sleq 2}}
$$
$smapsto textbf{1}_{{0leq s leq 2} }$ is a measurable function because it's a linear combination of indicators of borel sets, $smapstosqrt{2pi}e^{s^2over 2}$ is a measurable function because it's continuous, and $g$ is a measurable function because it's a composition of measurable functions. Thus,
$$
\ mathbb{E}(g(X))=int_{-infty}^infty g(s)f_X(s)ds=int_0^2ds=2
$$
answered Jan 19 at 15:20
J. DoeJ. Doe
15210
$endgroup$
We know that the density function of $X$ is
$f_X(s)={1oversqrt{2pi}}e^{-s^2over 2}$. Let $g:mathbb{R}tomathbb{R}$ a function defined by
$$
\ g(s)=sqrt{2pi}e^{s^2over 2}cdottextbf{1}_{{0leq sleq 2}}
$$
$smapsto textbf{1}_{{0leq s leq 2} }$ is a measurable function because it's a linear combination of indicators of borel sets, $smapstosqrt{2pi}e^{s^2over 2}$ is a measurable function because it's continuous, and $g$ is a measurable function because it's a composition of measurable functions. Thus,
$$
\ mathbb{E}(g(X))=int_{-infty}^infty g(s)f_X(s)ds=int_0^2ds=2
$$
answered Jan 19 at 15:20
J. DoeJ. Doe
15210
edited Jan 19 at 15:25
answered Jan 19 at 15:20
J. DoeJ. Doe
15210
answered Jan 19 at 15:20
J. DoeJ. Doe
15210
answered Jan 19 at 15:20
J. DoeJ. Doe
15210
15210
1
$begingroup$
This is fine and dandy but far too complicated, maybe? What if $g(x) = x+2$ for all $xin mathbb R$? Is $g(x)$ measurable? What is $E[g(X)] = E[X+2]$?
$endgroup$
– Dilip Sarwate
Jan 19 at 16:22
add a comment |
1
$begingroup$
This is fine and dandy but far too complicated, maybe? What if $g(x) = x+2$ for all $xin mathbb R$? Is $g(x)$ measurable? What is $E[g(X)] = E[X+2]$?
$endgroup$
– Dilip Sarwate
Jan 19 at 16:22
1
1
$begingroup$
This is fine and dandy but far too complicated, maybe? What if $g(x) = x+2$ for all $xin mathbb R$? Is $g(x)$ measurable? What is $E[g(X)] = E[X+2]$?
$endgroup$
– Dilip Sarwate
Jan 19 at 16:22
$begingroup$
This is fine and dandy but far too complicated, maybe? What if $g(x) = x+2$ for all $xin mathbb R$? Is $g(x)$ measurable? What is $E[g(X)] = E[X+2]$?
$endgroup$
– Dilip Sarwate
Jan 19 at 16:22
add a comment |
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$begingroup$
Hint: What is E(X^2)? Simpler: What is E(X)? Even simpler: What is E(1)?
$endgroup$
– Did
Jan 19 at 16:33