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Conditional distribution of joint normal when condition is an inequality
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Let $(varepsilon, nu)$ be jointly normally distributed:
$$beginpmatrixvarepsilon \ nu endpmatrix sim Nleft(beginpmatrix0 \ 0 endpmatrix , beginpmatrix sigma_varepsilon^2 & rho sigma_varepsilon \ rho sigma_varepsilon & 1 endpmatrix right)$$
where $rho$ is the correlation between $varepsilon$ and $nu$. Then show that for any real constant $alpha$:
$$varepsilon mid nu > alpha sim N(rho sigma_varepsilonlambda(-alpha), sigma_varepsilon^2(1-rho^2))$$
where $lambda(cdot) = fracphi(cdot)Phi(cdot)$, and $phi$ and $Phi$ are the pdf and cdf of a standard normal distribution, respectively.
I know that the conditional distribution of $varepsilon mid nu = alpha$ is normally distributed with mean $rhosigma_varepsilonalpha$ and variance $sigma_varepsilon^2(1-rho^2)$ by the properties of a joint normal distribution, but how do I prove the above when the condition is an inequality?
probability probability-distributions normal-distribution
asked Aug 6 at 1:16
elbarto
1,519523
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up vote
2
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Let $(varepsilon, nu)$ be jointly normally distributed:
$$beginpmatrixvarepsilon \ nu endpmatrix sim Nleft(beginpmatrix0 \ 0 endpmatrix , beginpmatrix sigma_varepsilon^2 & rho sigma_varepsilon \ rho sigma_varepsilon & 1 endpmatrix right)$$
where $rho$ is the correlation between $varepsilon$ and $nu$. Then show that for any real constant $alpha$:
$$varepsilon mid nu > alpha sim N(rho sigma_varepsilonlambda(-alpha), sigma_varepsilon^2(1-rho^2))$$
where $lambda(cdot) = fracphi(cdot)Phi(cdot)$, and $phi$ and $Phi$ are the pdf and cdf of a standard normal distribution, respectively.
I know that the conditional distribution of $varepsilon mid nu = alpha$ is normally distributed with mean $rhosigma_varepsilonalpha$ and variance $sigma_varepsilon^2(1-rho^2)$ by the properties of a joint normal distribution, but how do I prove the above when the condition is an inequality?
probability probability-distributions normal-distribution
asked Aug 6 at 1:16
elbarto
1,519523
You need the conditional density of $varepsilon vert nu > alpha$. This is given by $$ frac11-F_nu (alpha) int_alpha^infty f_varepsilon,nu(x,y) ,mathrm d y. $$ Here, $f_varepsilon,nu$ is the joint density, and $F_nu (alpha)$ is the (marginal) CDF of $nu$.
– Theoretical Economist
Aug 6 at 1:22
You could find the condition cdf $P(varepsilon le xmid nu > alpha)$ and see if it can be written in terms of the standard normal cdf $Phi$.
– StubbornAtom
Aug 6 at 7:19
Since $P(varepsilon le xmid nu >alpha)=fracP(varepsilon le x, nu >alpha)1-Phi(alpha)$, one would have to evaluate $$int_alpha^inftyint_-infty^x f_varepsilon,nu(varepsilon,nu),dvarepsilon,dnu$$ for the numerator. As per your result, the conditional cdf should evaluate to $Phileft(fracx-cdotcdotright)$.
– StubbornAtom
Aug 6 at 7:46
Similar question: math.stackexchange.com/questions/699457/….
– StubbornAtom
Aug 6 at 18:55
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $(varepsilon, nu)$ be jointly normally distributed:
$$beginpmatrixvarepsilon \ nu endpmatrix sim Nleft(beginpmatrix0 \ 0 endpmatrix , beginpmatrix sigma_varepsilon^2 & rho sigma_varepsilon \ rho sigma_varepsilon & 1 endpmatrix right)$$
where $rho$ is the correlation between $varepsilon$ and $nu$. Then show that for any real constant $alpha$:
$$varepsilon mid nu > alpha sim N(rho sigma_varepsilonlambda(-alpha), sigma_varepsilon^2(1-rho^2))$$
where $lambda(cdot) = fracphi(cdot)Phi(cdot)$, and $phi$ and $Phi$ are the pdf and cdf of a standard normal distribution, respectively.
I know that the conditional distribution of $varepsilon mid nu = alpha$ is normally distributed with mean $rhosigma_varepsilonalpha$ and variance $sigma_varepsilon^2(1-rho^2)$ by the properties of a joint normal distribution, but how do I prove the above when the condition is an inequality?
probability probability-distributions normal-distribution
asked Aug 6 at 1:16
elbarto
1,519523
Let $(varepsilon, nu)$ be jointly normally distributed:
$$beginpmatrixvarepsilon \ nu endpmatrix sim Nleft(beginpmatrix0 \ 0 endpmatrix , beginpmatrix sigma_varepsilon^2 & rho sigma_varepsilon \ rho sigma_varepsilon & 1 endpmatrix right)$$
where $rho$ is the correlation between $varepsilon$ and $nu$. Then show that for any real constant $alpha$:
$$varepsilon mid nu > alpha sim N(rho sigma_varepsilonlambda(-alpha), sigma_varepsilon^2(1-rho^2))$$
where $lambda(cdot) = fracphi(cdot)Phi(cdot)$, and $phi$ and $Phi$ are the pdf and cdf of a standard normal distribution, respectively.
I know that the conditional distribution of $varepsilon mid nu = alpha$ is normally distributed with mean $rhosigma_varepsilonalpha$ and variance $sigma_varepsilon^2(1-rho^2)$ by the properties of a joint normal distribution, but how do I prove the above when the condition is an inequality?
probability probability-distributions normal-distribution
asked Aug 6 at 1:16
elbarto
1,519523
asked Aug 6 at 1:16
elbarto
1,519523
asked Aug 6 at 1:16
elbarto
1,519523
asked Aug 6 at 1:16
elbarto
1,519523
1,519523
You need the conditional density of $varepsilon vert nu > alpha$. This is given by $$ frac11-F_nu (alpha) int_alpha^infty f_varepsilon,nu(x,y) ,mathrm d y. $$ Here, $f_varepsilon,nu$ is the joint density, and $F_nu (alpha)$ is the (marginal) CDF of $nu$.
– Theoretical Economist
Aug 6 at 1:22
You could find the condition cdf $P(varepsilon le xmid nu > alpha)$ and see if it can be written in terms of the standard normal cdf $Phi$.
– StubbornAtom
Aug 6 at 7:19
Since $P(varepsilon le xmid nu >alpha)=fracP(varepsilon le x, nu >alpha)1-Phi(alpha)$, one would have to evaluate $$int_alpha^inftyint_-infty^x f_varepsilon,nu(varepsilon,nu),dvarepsilon,dnu$$ for the numerator. As per your result, the conditional cdf should evaluate to $Phileft(fracx-cdotcdotright)$.
– StubbornAtom
Aug 6 at 7:46
Similar question: math.stackexchange.com/questions/699457/….
– StubbornAtom
Aug 6 at 18:55
add a comment |Â
You need the conditional density of $varepsilon vert nu > alpha$. This is given by $$ frac11-F_nu (alpha) int_alpha^infty f_varepsilon,nu(x,y) ,mathrm d y. $$ Here, $f_varepsilon,nu$ is the joint density, and $F_nu (alpha)$ is the (marginal) CDF of $nu$.
– Theoretical Economist
Aug 6 at 1:22
You could find the condition cdf $P(varepsilon le xmid nu > alpha)$ and see if it can be written in terms of the standard normal cdf $Phi$.
– StubbornAtom
Aug 6 at 7:19
Since $P(varepsilon le xmid nu >alpha)=fracP(varepsilon le x, nu >alpha)1-Phi(alpha)$, one would have to evaluate $$int_alpha^inftyint_-infty^x f_varepsilon,nu(varepsilon,nu),dvarepsilon,dnu$$ for the numerator. As per your result, the conditional cdf should evaluate to $Phileft(fracx-cdotcdotright)$.
– StubbornAtom
Aug 6 at 7:46
Similar question: math.stackexchange.com/questions/699457/….
– StubbornAtom
Aug 6 at 18:55
You need the conditional density of $varepsilon vert nu > alpha$. This is given by $$ frac11-F_nu (alpha) int_alpha^infty f_varepsilon,nu(x,y) ,mathrm d y. $$ Here, $f_varepsilon,nu$ is the joint density, and $F_nu (alpha)$ is the (marginal) CDF of $nu$.
– Theoretical Economist
Aug 6 at 1:22
You need the conditional density of $varepsilon vert nu > alpha$. This is given by $$ frac11-F_nu (alpha) int_alpha^infty f_varepsilon,nu(x,y) ,mathrm d y. $$ Here, $f_varepsilon,nu$ is the joint density, and $F_nu (alpha)$ is the (marginal) CDF of $nu$.
– Theoretical Economist
Aug 6 at 1:22
You could find the condition cdf $P(varepsilon le xmid nu > alpha)$ and see if it can be written in terms of the standard normal cdf $Phi$.
– StubbornAtom
Aug 6 at 7:19
You could find the condition cdf $P(varepsilon le xmid nu > alpha)$ and see if it can be written in terms of the standard normal cdf $Phi$.
– StubbornAtom
Aug 6 at 7:19
Since $P(varepsilon le xmid nu >alpha)=fracP(varepsilon le x, nu >alpha)1-Phi(alpha)$, one would have to evaluate $$int_alpha^inftyint_-infty^x f_varepsilon,nu(varepsilon,nu),dvarepsilon,dnu$$ for the numerator. As per your result, the conditional cdf should evaluate to $Phileft(fracx-cdotcdotright)$.
– StubbornAtom
Aug 6 at 7:46
Since $P(varepsilon le xmid nu >alpha)=fracP(varepsilon le x, nu >alpha)1-Phi(alpha)$, one would have to evaluate $$int_alpha^inftyint_-infty^x f_varepsilon,nu(varepsilon,nu),dvarepsilon,dnu$$ for the numerator. As per your result, the conditional cdf should evaluate to $Phileft(fracx-cdotcdotright)$.
– StubbornAtom
Aug 6 at 7:46
Similar question: math.stackexchange.com/questions/699457/….
– StubbornAtom
Aug 6 at 18:55
Similar question: math.stackexchange.com/questions/699457/….
– StubbornAtom
Aug 6 at 18:55
add a comment |Â
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You need the conditional density of $varepsilon vert nu > alpha$. This is given by $$ frac11-F_nu (alpha) int_alpha^infty f_varepsilon,nu(x,y) ,mathrm d y. $$ Here, $f_varepsilon,nu$ is the joint density, and $F_nu (alpha)$ is the (marginal) CDF of $nu$.
– Theoretical Economist
Aug 6 at 1:22
You could find the condition cdf $P(varepsilon le xmid nu > alpha)$ and see if it can be written in terms of the standard normal cdf $Phi$.
– StubbornAtom
Aug 6 at 7:19
Since $P(varepsilon le xmid nu >alpha)=fracP(varepsilon le x, nu >alpha)1-Phi(alpha)$, one would have to evaluate $$int_alpha^inftyint_-infty^x f_varepsilon,nu(varepsilon,nu),dvarepsilon,dnu$$ for the numerator. As per your result, the conditional cdf should evaluate to $Phileft(fracx-cdotcdotright)$.
– StubbornAtom
Aug 6 at 7:46
Similar question: math.stackexchange.com/questions/699457/….
– StubbornAtom
Aug 6 at 18:55