$P(X>0,Y>0)$ for a bivariate normal distribution with correlation $rho$
$begingroup$
$X$ and $Y$ have a bivariate normal distribution with $rho$ as covariance. $X$ and $Y$ are standard normal variables.
I showed that $X$ and $Z= {frac{Y-rho X}{sqrt{(1-rho^2)}}}$ are independent standard normal variables.
Using this I need to show that
$$P(X >0,Y>0) = frac14 + frac{1}{2pi} cdotmathrm{arcsin}(rho).$$
probability probability-distributions normal-distribution bivariate-distributions
$endgroup$
add a comment |
$begingroup$
$X$ and $Y$ have a bivariate normal distribution with $rho$ as covariance. $X$ and $Y$ are standard normal variables.
I showed that $X$ and $Z= {frac{Y-rho X}{sqrt{(1-rho^2)}}}$ are independent standard normal variables.
Using this I need to show that
$$P(X >0,Y>0) = frac14 + frac{1}{2pi} cdotmathrm{arcsin}(rho).$$
probability probability-distributions normal-distribution bivariate-distributions
$endgroup$
$begingroup$
As a hint for the first question, notice that $Z$ is the conditional of $Y$ on $X$.
$endgroup$
– Learner
Dec 10 '12 at 11:03
$begingroup$
it is easy to solve part. As X and Y are standard normal variables. But I am stuck at part two.
$endgroup$
– user669083
Dec 10 '12 at 11:10
$begingroup$
If it is easy for part 1, why are you asking the question?
$endgroup$
– Learner
Dec 10 '12 at 11:10
$begingroup$
Because the second part is to use first part to solve second.
$endgroup$
– user669083
Dec 10 '12 at 11:12
$begingroup$
@user669083 Learner is right. If you want an answer for part II, then you shouldnt say: "I had to show..." You need to say "using the fact that .....", I need to show : "part II"
$endgroup$
– Seyhmus Güngören
Dec 10 '12 at 12:23
add a comment |
$begingroup$
$X$ and $Y$ have a bivariate normal distribution with $rho$ as covariance. $X$ and $Y$ are standard normal variables.
I showed that $X$ and $Z= {frac{Y-rho X}{sqrt{(1-rho^2)}}}$ are independent standard normal variables.
Using this I need to show that
$$P(X >0,Y>0) = frac14 + frac{1}{2pi} cdotmathrm{arcsin}(rho).$$
probability probability-distributions normal-distribution bivariate-distributions
$endgroup$
$X$ and $Y$ have a bivariate normal distribution with $rho$ as covariance. $X$ and $Y$ are standard normal variables.
I showed that $X$ and $Z= {frac{Y-rho X}{sqrt{(1-rho^2)}}}$ are independent standard normal variables.
Using this I need to show that
$$P(X >0,Y>0) = frac14 + frac{1}{2pi} cdotmathrm{arcsin}(rho).$$
probability probability-distributions normal-distribution bivariate-distributions
probability probability-distributions normal-distribution bivariate-distributions
edited Dec 12 '18 at 22:56
grand_chat
20.4k11327
20.4k11327
asked Dec 10 '12 at 10:51
user669083user669083
4312920
4312920
$begingroup$
As a hint for the first question, notice that $Z$ is the conditional of $Y$ on $X$.
$endgroup$
– Learner
Dec 10 '12 at 11:03
$begingroup$
it is easy to solve part. As X and Y are standard normal variables. But I am stuck at part two.
$endgroup$
– user669083
Dec 10 '12 at 11:10
$begingroup$
If it is easy for part 1, why are you asking the question?
$endgroup$
– Learner
Dec 10 '12 at 11:10
$begingroup$
Because the second part is to use first part to solve second.
$endgroup$
– user669083
Dec 10 '12 at 11:12
$begingroup$
@user669083 Learner is right. If you want an answer for part II, then you shouldnt say: "I had to show..." You need to say "using the fact that .....", I need to show : "part II"
$endgroup$
– Seyhmus Güngören
Dec 10 '12 at 12:23
add a comment |
$begingroup$
As a hint for the first question, notice that $Z$ is the conditional of $Y$ on $X$.
$endgroup$
– Learner
Dec 10 '12 at 11:03
$begingroup$
it is easy to solve part. As X and Y are standard normal variables. But I am stuck at part two.
$endgroup$
– user669083
Dec 10 '12 at 11:10
$begingroup$
If it is easy for part 1, why are you asking the question?
$endgroup$
– Learner
Dec 10 '12 at 11:10
$begingroup$
Because the second part is to use first part to solve second.
$endgroup$
– user669083
Dec 10 '12 at 11:12
$begingroup$
@user669083 Learner is right. If you want an answer for part II, then you shouldnt say: "I had to show..." You need to say "using the fact that .....", I need to show : "part II"
$endgroup$
– Seyhmus Güngören
Dec 10 '12 at 12:23
$begingroup$
As a hint for the first question, notice that $Z$ is the conditional of $Y$ on $X$.
$endgroup$
– Learner
Dec 10 '12 at 11:03
$begingroup$
As a hint for the first question, notice that $Z$ is the conditional of $Y$ on $X$.
$endgroup$
– Learner
Dec 10 '12 at 11:03
$begingroup$
it is easy to solve part. As X and Y are standard normal variables. But I am stuck at part two.
$endgroup$
– user669083
Dec 10 '12 at 11:10
$begingroup$
it is easy to solve part. As X and Y are standard normal variables. But I am stuck at part two.
$endgroup$
– user669083
Dec 10 '12 at 11:10
$begingroup$
If it is easy for part 1, why are you asking the question?
$endgroup$
– Learner
Dec 10 '12 at 11:10
$begingroup$
If it is easy for part 1, why are you asking the question?
$endgroup$
– Learner
Dec 10 '12 at 11:10
$begingroup$
Because the second part is to use first part to solve second.
$endgroup$
– user669083
Dec 10 '12 at 11:12
$begingroup$
Because the second part is to use first part to solve second.
$endgroup$
– user669083
Dec 10 '12 at 11:12
$begingroup$
@user669083 Learner is right. If you want an answer for part II, then you shouldnt say: "I had to show..." You need to say "using the fact that .....", I need to show : "part II"
$endgroup$
– Seyhmus Güngören
Dec 10 '12 at 12:23
$begingroup$
@user669083 Learner is right. If you want an answer for part II, then you shouldnt say: "I had to show..." You need to say "using the fact that .....", I need to show : "part II"
$endgroup$
– Seyhmus Güngören
Dec 10 '12 at 12:23
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Let $left( V, W right)$ be i.i.d standard normal, let $U$ be uniform on
$left[ - pi, pi right)$ independent of $R = sqrt{V^2 + W^2}$ and let
$varphi = arcsin rho$. Then the following vectors have the same
distributions (think about the Box-Muller transformation)
begin{eqnarray*}
left( X, Y right) & overset{d}{=} & sqrt{2} left( V cos varphi + W
sin varphi, W right)\
left( V, W right) & overset{d}{=} & R left( cos U, sin U right)
end{eqnarray*}
Implying
begin{eqnarray*}
P left[ X > 0, Y > 0 right] & = & P left[ cos U cos varphi + sin U
sin varphi > 0, sin U > 0 right]\
& = & P left[ U in left( varphi - frac{pi}{2}, varphi +
frac{pi}{2} right) cap left( 0, pi right) right]\
& = & frac{varphi}{2 pi} + frac{1}{4}
end{eqnarray*}
Here is an alternative proof:
Let $phi$ be the density of the standard normal distribution
begin{eqnarray*}
P left[ X > 0, Y > 0 right] & = & P left[ X < 0, Y < 0 right]\
& = & int_{- infty}^0 phi left( z right) int_{- infty}^0
frac{1}{sqrt{1 - rho^2}} phi left( frac{x - rho z}{sqrt{1 - rho^2}}
right) mathrm{d} x mathrm{d} z\
& = & int_{- infty}^0 phi left( z right) int_{- infty}^{- frac{rho
z}{sqrt{1 - rho^2}}} phi left( x right) mathrm{d} x mathrm{d} z
end{eqnarray*}
Let's call the above integral $h left( rho right)$, then after some
simplifications
$$frac{partial h left( rho right)}{partial rho} = frac{1}{2 pi
sqrt{1 - rho^2}} $$
By integrating back (or considering the problem as a first-order ordinary differential equation), $h left( rho right) = frac{1}{2 pi} arcsin rho +
K$ where $K$ is some constant. By the special case of independence, $h left( 0 right) =
frac{1}{4}$, you get the final solution
$$ P left[ X > 0, Y > 0 right] = frac{1}{2 pi} arcsin rho + frac{1}{4}$$
$endgroup$
$begingroup$
is there a simpler solution ?
$endgroup$
– user669083
Dec 10 '12 at 17:58
$begingroup$
@user669083 If you do not understand the above proof, you should point to the part where you have problems. Anyway, I am going to add an alternative proof (I am not sure it is simpler).
$endgroup$
– Learner
Dec 11 '12 at 1:08
$begingroup$
Let's call the above integral h(ρ), then after some simplifications. What is your simplification? Thanks in advance.
$endgroup$
– user52613
Dec 11 '12 at 9:38
$begingroup$
@Fafsads Please check en.wikipedia.org/wiki/Leibniz_integral_rule for how to differentiate with respect to $rho$. After that, all is needed is some simple manipulations of one-dimensional normal density integrals. Let me know after that if you are still having trouble.
$endgroup$
– Learner
Dec 11 '12 at 10:34
1
$begingroup$
I solved it by writing the equation as $(PX>0,Z> {frac{-rho X}{sqrt{(1-rho^2)}}})$. After that write in terms of integral, convert into polar co-ordinates and voila I got the answer
$endgroup$
– user669083
Dec 16 '12 at 9:52
add a comment |
$begingroup$
Using the OP's hint: The event ${X>0,Y>0}$ is the same as the event ${X>0,Z>frac{-rho}{sqrt{1-rho^2}}X}$, where now $X$ and $Z$ are independent standard normal variables. Writing $a:=frac{-rho}{sqrt{1-rho^2}}$ for brevity, the desired probability is expressible as a double integral involving the joint density of $(X,Z)$:
$$
P(X>0,Y>0)=P(X>0,Z>aX)=int_{x=0}^inftyint_{z=ax}^inftyfrac1{sqrt{2pi}}e^{-x^2/2}frac1{sqrt{2pi}}e^{-z^2/2},dz,dx.
$$
Switching to polar coordinates, this equals
$$
begin{align}
int_{theta=arctan(a)}^{pi/2}int_{r=0}^inftyfrac1{2pi}e^{-r^2/2},rdrdtheta&=int_{theta=arctan(a)}^{pi/2}frac1{2pi},dtheta\
&=frac1{2pi}left(fracpi2-arctan aright)\
&=frac14+frac1{2pi}arctanfracrho{sqrt{1-rho^2}};
end{align}
$$
for the last equality we substitute $a:=frac{-rho}{sqrt{1-rho^2}}$ and use the fact that the arctan function is odd. To finish off, remember that if $thetain[-pi/2,pi/2]$ then $$theta=arctanfrac{rho}{sqrt{1-rho^2}} Longleftrightarrow tantheta=frac{rho}{sqrt{1-rho^2}} Longleftrightarrow sintheta=rho Longleftrightarrow theta=arcsin rho.$$
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f255368%2fpx0-y0-for-a-bivariate-normal-distribution-with-correlation-rho%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Let $left( V, W right)$ be i.i.d standard normal, let $U$ be uniform on
$left[ - pi, pi right)$ independent of $R = sqrt{V^2 + W^2}$ and let
$varphi = arcsin rho$. Then the following vectors have the same
distributions (think about the Box-Muller transformation)
begin{eqnarray*}
left( X, Y right) & overset{d}{=} & sqrt{2} left( V cos varphi + W
sin varphi, W right)\
left( V, W right) & overset{d}{=} & R left( cos U, sin U right)
end{eqnarray*}
Implying
begin{eqnarray*}
P left[ X > 0, Y > 0 right] & = & P left[ cos U cos varphi + sin U
sin varphi > 0, sin U > 0 right]\
& = & P left[ U in left( varphi - frac{pi}{2}, varphi +
frac{pi}{2} right) cap left( 0, pi right) right]\
& = & frac{varphi}{2 pi} + frac{1}{4}
end{eqnarray*}
Here is an alternative proof:
Let $phi$ be the density of the standard normal distribution
begin{eqnarray*}
P left[ X > 0, Y > 0 right] & = & P left[ X < 0, Y < 0 right]\
& = & int_{- infty}^0 phi left( z right) int_{- infty}^0
frac{1}{sqrt{1 - rho^2}} phi left( frac{x - rho z}{sqrt{1 - rho^2}}
right) mathrm{d} x mathrm{d} z\
& = & int_{- infty}^0 phi left( z right) int_{- infty}^{- frac{rho
z}{sqrt{1 - rho^2}}} phi left( x right) mathrm{d} x mathrm{d} z
end{eqnarray*}
Let's call the above integral $h left( rho right)$, then after some
simplifications
$$frac{partial h left( rho right)}{partial rho} = frac{1}{2 pi
sqrt{1 - rho^2}} $$
By integrating back (or considering the problem as a first-order ordinary differential equation), $h left( rho right) = frac{1}{2 pi} arcsin rho +
K$ where $K$ is some constant. By the special case of independence, $h left( 0 right) =
frac{1}{4}$, you get the final solution
$$ P left[ X > 0, Y > 0 right] = frac{1}{2 pi} arcsin rho + frac{1}{4}$$
$endgroup$
$begingroup$
is there a simpler solution ?
$endgroup$
– user669083
Dec 10 '12 at 17:58
$begingroup$
@user669083 If you do not understand the above proof, you should point to the part where you have problems. Anyway, I am going to add an alternative proof (I am not sure it is simpler).
$endgroup$
– Learner
Dec 11 '12 at 1:08
$begingroup$
Let's call the above integral h(ρ), then after some simplifications. What is your simplification? Thanks in advance.
$endgroup$
– user52613
Dec 11 '12 at 9:38
$begingroup$
@Fafsads Please check en.wikipedia.org/wiki/Leibniz_integral_rule for how to differentiate with respect to $rho$. After that, all is needed is some simple manipulations of one-dimensional normal density integrals. Let me know after that if you are still having trouble.
$endgroup$
– Learner
Dec 11 '12 at 10:34
1
$begingroup$
I solved it by writing the equation as $(PX>0,Z> {frac{-rho X}{sqrt{(1-rho^2)}}})$. After that write in terms of integral, convert into polar co-ordinates and voila I got the answer
$endgroup$
– user669083
Dec 16 '12 at 9:52
add a comment |
$begingroup$
Let $left( V, W right)$ be i.i.d standard normal, let $U$ be uniform on
$left[ - pi, pi right)$ independent of $R = sqrt{V^2 + W^2}$ and let
$varphi = arcsin rho$. Then the following vectors have the same
distributions (think about the Box-Muller transformation)
begin{eqnarray*}
left( X, Y right) & overset{d}{=} & sqrt{2} left( V cos varphi + W
sin varphi, W right)\
left( V, W right) & overset{d}{=} & R left( cos U, sin U right)
end{eqnarray*}
Implying
begin{eqnarray*}
P left[ X > 0, Y > 0 right] & = & P left[ cos U cos varphi + sin U
sin varphi > 0, sin U > 0 right]\
& = & P left[ U in left( varphi - frac{pi}{2}, varphi +
frac{pi}{2} right) cap left( 0, pi right) right]\
& = & frac{varphi}{2 pi} + frac{1}{4}
end{eqnarray*}
Here is an alternative proof:
Let $phi$ be the density of the standard normal distribution
begin{eqnarray*}
P left[ X > 0, Y > 0 right] & = & P left[ X < 0, Y < 0 right]\
& = & int_{- infty}^0 phi left( z right) int_{- infty}^0
frac{1}{sqrt{1 - rho^2}} phi left( frac{x - rho z}{sqrt{1 - rho^2}}
right) mathrm{d} x mathrm{d} z\
& = & int_{- infty}^0 phi left( z right) int_{- infty}^{- frac{rho
z}{sqrt{1 - rho^2}}} phi left( x right) mathrm{d} x mathrm{d} z
end{eqnarray*}
Let's call the above integral $h left( rho right)$, then after some
simplifications
$$frac{partial h left( rho right)}{partial rho} = frac{1}{2 pi
sqrt{1 - rho^2}} $$
By integrating back (or considering the problem as a first-order ordinary differential equation), $h left( rho right) = frac{1}{2 pi} arcsin rho +
K$ where $K$ is some constant. By the special case of independence, $h left( 0 right) =
frac{1}{4}$, you get the final solution
$$ P left[ X > 0, Y > 0 right] = frac{1}{2 pi} arcsin rho + frac{1}{4}$$
$endgroup$
$begingroup$
is there a simpler solution ?
$endgroup$
– user669083
Dec 10 '12 at 17:58
$begingroup$
@user669083 If you do not understand the above proof, you should point to the part where you have problems. Anyway, I am going to add an alternative proof (I am not sure it is simpler).
$endgroup$
– Learner
Dec 11 '12 at 1:08
$begingroup$
Let's call the above integral h(ρ), then after some simplifications. What is your simplification? Thanks in advance.
$endgroup$
– user52613
Dec 11 '12 at 9:38
$begingroup$
@Fafsads Please check en.wikipedia.org/wiki/Leibniz_integral_rule for how to differentiate with respect to $rho$. After that, all is needed is some simple manipulations of one-dimensional normal density integrals. Let me know after that if you are still having trouble.
$endgroup$
– Learner
Dec 11 '12 at 10:34
1
$begingroup$
I solved it by writing the equation as $(PX>0,Z> {frac{-rho X}{sqrt{(1-rho^2)}}})$. After that write in terms of integral, convert into polar co-ordinates and voila I got the answer
$endgroup$
– user669083
Dec 16 '12 at 9:52
add a comment |
$begingroup$
Let $left( V, W right)$ be i.i.d standard normal, let $U$ be uniform on
$left[ - pi, pi right)$ independent of $R = sqrt{V^2 + W^2}$ and let
$varphi = arcsin rho$. Then the following vectors have the same
distributions (think about the Box-Muller transformation)
begin{eqnarray*}
left( X, Y right) & overset{d}{=} & sqrt{2} left( V cos varphi + W
sin varphi, W right)\
left( V, W right) & overset{d}{=} & R left( cos U, sin U right)
end{eqnarray*}
Implying
begin{eqnarray*}
P left[ X > 0, Y > 0 right] & = & P left[ cos U cos varphi + sin U
sin varphi > 0, sin U > 0 right]\
& = & P left[ U in left( varphi - frac{pi}{2}, varphi +
frac{pi}{2} right) cap left( 0, pi right) right]\
& = & frac{varphi}{2 pi} + frac{1}{4}
end{eqnarray*}
Here is an alternative proof:
Let $phi$ be the density of the standard normal distribution
begin{eqnarray*}
P left[ X > 0, Y > 0 right] & = & P left[ X < 0, Y < 0 right]\
& = & int_{- infty}^0 phi left( z right) int_{- infty}^0
frac{1}{sqrt{1 - rho^2}} phi left( frac{x - rho z}{sqrt{1 - rho^2}}
right) mathrm{d} x mathrm{d} z\
& = & int_{- infty}^0 phi left( z right) int_{- infty}^{- frac{rho
z}{sqrt{1 - rho^2}}} phi left( x right) mathrm{d} x mathrm{d} z
end{eqnarray*}
Let's call the above integral $h left( rho right)$, then after some
simplifications
$$frac{partial h left( rho right)}{partial rho} = frac{1}{2 pi
sqrt{1 - rho^2}} $$
By integrating back (or considering the problem as a first-order ordinary differential equation), $h left( rho right) = frac{1}{2 pi} arcsin rho +
K$ where $K$ is some constant. By the special case of independence, $h left( 0 right) =
frac{1}{4}$, you get the final solution
$$ P left[ X > 0, Y > 0 right] = frac{1}{2 pi} arcsin rho + frac{1}{4}$$
$endgroup$
Let $left( V, W right)$ be i.i.d standard normal, let $U$ be uniform on
$left[ - pi, pi right)$ independent of $R = sqrt{V^2 + W^2}$ and let
$varphi = arcsin rho$. Then the following vectors have the same
distributions (think about the Box-Muller transformation)
begin{eqnarray*}
left( X, Y right) & overset{d}{=} & sqrt{2} left( V cos varphi + W
sin varphi, W right)\
left( V, W right) & overset{d}{=} & R left( cos U, sin U right)
end{eqnarray*}
Implying
begin{eqnarray*}
P left[ X > 0, Y > 0 right] & = & P left[ cos U cos varphi + sin U
sin varphi > 0, sin U > 0 right]\
& = & P left[ U in left( varphi - frac{pi}{2}, varphi +
frac{pi}{2} right) cap left( 0, pi right) right]\
& = & frac{varphi}{2 pi} + frac{1}{4}
end{eqnarray*}
Here is an alternative proof:
Let $phi$ be the density of the standard normal distribution
begin{eqnarray*}
P left[ X > 0, Y > 0 right] & = & P left[ X < 0, Y < 0 right]\
& = & int_{- infty}^0 phi left( z right) int_{- infty}^0
frac{1}{sqrt{1 - rho^2}} phi left( frac{x - rho z}{sqrt{1 - rho^2}}
right) mathrm{d} x mathrm{d} z\
& = & int_{- infty}^0 phi left( z right) int_{- infty}^{- frac{rho
z}{sqrt{1 - rho^2}}} phi left( x right) mathrm{d} x mathrm{d} z
end{eqnarray*}
Let's call the above integral $h left( rho right)$, then after some
simplifications
$$frac{partial h left( rho right)}{partial rho} = frac{1}{2 pi
sqrt{1 - rho^2}} $$
By integrating back (or considering the problem as a first-order ordinary differential equation), $h left( rho right) = frac{1}{2 pi} arcsin rho +
K$ where $K$ is some constant. By the special case of independence, $h left( 0 right) =
frac{1}{4}$, you get the final solution
$$ P left[ X > 0, Y > 0 right] = frac{1}{2 pi} arcsin rho + frac{1}{4}$$
edited Dec 11 '12 at 12:45
answered Dec 10 '12 at 12:01
LearnerLearner
5,08022034
5,08022034
$begingroup$
is there a simpler solution ?
$endgroup$
– user669083
Dec 10 '12 at 17:58
$begingroup$
@user669083 If you do not understand the above proof, you should point to the part where you have problems. Anyway, I am going to add an alternative proof (I am not sure it is simpler).
$endgroup$
– Learner
Dec 11 '12 at 1:08
$begingroup$
Let's call the above integral h(ρ), then after some simplifications. What is your simplification? Thanks in advance.
$endgroup$
– user52613
Dec 11 '12 at 9:38
$begingroup$
@Fafsads Please check en.wikipedia.org/wiki/Leibniz_integral_rule for how to differentiate with respect to $rho$. After that, all is needed is some simple manipulations of one-dimensional normal density integrals. Let me know after that if you are still having trouble.
$endgroup$
– Learner
Dec 11 '12 at 10:34
1
$begingroup$
I solved it by writing the equation as $(PX>0,Z> {frac{-rho X}{sqrt{(1-rho^2)}}})$. After that write in terms of integral, convert into polar co-ordinates and voila I got the answer
$endgroup$
– user669083
Dec 16 '12 at 9:52
add a comment |
$begingroup$
is there a simpler solution ?
$endgroup$
– user669083
Dec 10 '12 at 17:58
$begingroup$
@user669083 If you do not understand the above proof, you should point to the part where you have problems. Anyway, I am going to add an alternative proof (I am not sure it is simpler).
$endgroup$
– Learner
Dec 11 '12 at 1:08
$begingroup$
Let's call the above integral h(ρ), then after some simplifications. What is your simplification? Thanks in advance.
$endgroup$
– user52613
Dec 11 '12 at 9:38
$begingroup$
@Fafsads Please check en.wikipedia.org/wiki/Leibniz_integral_rule for how to differentiate with respect to $rho$. After that, all is needed is some simple manipulations of one-dimensional normal density integrals. Let me know after that if you are still having trouble.
$endgroup$
– Learner
Dec 11 '12 at 10:34
1
$begingroup$
I solved it by writing the equation as $(PX>0,Z> {frac{-rho X}{sqrt{(1-rho^2)}}})$. After that write in terms of integral, convert into polar co-ordinates and voila I got the answer
$endgroup$
– user669083
Dec 16 '12 at 9:52
$begingroup$
is there a simpler solution ?
$endgroup$
– user669083
Dec 10 '12 at 17:58
$begingroup$
is there a simpler solution ?
$endgroup$
– user669083
Dec 10 '12 at 17:58
$begingroup$
@user669083 If you do not understand the above proof, you should point to the part where you have problems. Anyway, I am going to add an alternative proof (I am not sure it is simpler).
$endgroup$
– Learner
Dec 11 '12 at 1:08
$begingroup$
@user669083 If you do not understand the above proof, you should point to the part where you have problems. Anyway, I am going to add an alternative proof (I am not sure it is simpler).
$endgroup$
– Learner
Dec 11 '12 at 1:08
$begingroup$
Let's call the above integral h(ρ), then after some simplifications. What is your simplification? Thanks in advance.
$endgroup$
– user52613
Dec 11 '12 at 9:38
$begingroup$
Let's call the above integral h(ρ), then after some simplifications. What is your simplification? Thanks in advance.
$endgroup$
– user52613
Dec 11 '12 at 9:38
$begingroup$
@Fafsads Please check en.wikipedia.org/wiki/Leibniz_integral_rule for how to differentiate with respect to $rho$. After that, all is needed is some simple manipulations of one-dimensional normal density integrals. Let me know after that if you are still having trouble.
$endgroup$
– Learner
Dec 11 '12 at 10:34
$begingroup$
@Fafsads Please check en.wikipedia.org/wiki/Leibniz_integral_rule for how to differentiate with respect to $rho$. After that, all is needed is some simple manipulations of one-dimensional normal density integrals. Let me know after that if you are still having trouble.
$endgroup$
– Learner
Dec 11 '12 at 10:34
1
1
$begingroup$
I solved it by writing the equation as $(PX>0,Z> {frac{-rho X}{sqrt{(1-rho^2)}}})$. After that write in terms of integral, convert into polar co-ordinates and voila I got the answer
$endgroup$
– user669083
Dec 16 '12 at 9:52
$begingroup$
I solved it by writing the equation as $(PX>0,Z> {frac{-rho X}{sqrt{(1-rho^2)}}})$. After that write in terms of integral, convert into polar co-ordinates and voila I got the answer
$endgroup$
– user669083
Dec 16 '12 at 9:52
add a comment |
$begingroup$
Using the OP's hint: The event ${X>0,Y>0}$ is the same as the event ${X>0,Z>frac{-rho}{sqrt{1-rho^2}}X}$, where now $X$ and $Z$ are independent standard normal variables. Writing $a:=frac{-rho}{sqrt{1-rho^2}}$ for brevity, the desired probability is expressible as a double integral involving the joint density of $(X,Z)$:
$$
P(X>0,Y>0)=P(X>0,Z>aX)=int_{x=0}^inftyint_{z=ax}^inftyfrac1{sqrt{2pi}}e^{-x^2/2}frac1{sqrt{2pi}}e^{-z^2/2},dz,dx.
$$
Switching to polar coordinates, this equals
$$
begin{align}
int_{theta=arctan(a)}^{pi/2}int_{r=0}^inftyfrac1{2pi}e^{-r^2/2},rdrdtheta&=int_{theta=arctan(a)}^{pi/2}frac1{2pi},dtheta\
&=frac1{2pi}left(fracpi2-arctan aright)\
&=frac14+frac1{2pi}arctanfracrho{sqrt{1-rho^2}};
end{align}
$$
for the last equality we substitute $a:=frac{-rho}{sqrt{1-rho^2}}$ and use the fact that the arctan function is odd. To finish off, remember that if $thetain[-pi/2,pi/2]$ then $$theta=arctanfrac{rho}{sqrt{1-rho^2}} Longleftrightarrow tantheta=frac{rho}{sqrt{1-rho^2}} Longleftrightarrow sintheta=rho Longleftrightarrow theta=arcsin rho.$$
$endgroup$
add a comment |
$begingroup$
Using the OP's hint: The event ${X>0,Y>0}$ is the same as the event ${X>0,Z>frac{-rho}{sqrt{1-rho^2}}X}$, where now $X$ and $Z$ are independent standard normal variables. Writing $a:=frac{-rho}{sqrt{1-rho^2}}$ for brevity, the desired probability is expressible as a double integral involving the joint density of $(X,Z)$:
$$
P(X>0,Y>0)=P(X>0,Z>aX)=int_{x=0}^inftyint_{z=ax}^inftyfrac1{sqrt{2pi}}e^{-x^2/2}frac1{sqrt{2pi}}e^{-z^2/2},dz,dx.
$$
Switching to polar coordinates, this equals
$$
begin{align}
int_{theta=arctan(a)}^{pi/2}int_{r=0}^inftyfrac1{2pi}e^{-r^2/2},rdrdtheta&=int_{theta=arctan(a)}^{pi/2}frac1{2pi},dtheta\
&=frac1{2pi}left(fracpi2-arctan aright)\
&=frac14+frac1{2pi}arctanfracrho{sqrt{1-rho^2}};
end{align}
$$
for the last equality we substitute $a:=frac{-rho}{sqrt{1-rho^2}}$ and use the fact that the arctan function is odd. To finish off, remember that if $thetain[-pi/2,pi/2]$ then $$theta=arctanfrac{rho}{sqrt{1-rho^2}} Longleftrightarrow tantheta=frac{rho}{sqrt{1-rho^2}} Longleftrightarrow sintheta=rho Longleftrightarrow theta=arcsin rho.$$
$endgroup$
add a comment |
$begingroup$
Using the OP's hint: The event ${X>0,Y>0}$ is the same as the event ${X>0,Z>frac{-rho}{sqrt{1-rho^2}}X}$, where now $X$ and $Z$ are independent standard normal variables. Writing $a:=frac{-rho}{sqrt{1-rho^2}}$ for brevity, the desired probability is expressible as a double integral involving the joint density of $(X,Z)$:
$$
P(X>0,Y>0)=P(X>0,Z>aX)=int_{x=0}^inftyint_{z=ax}^inftyfrac1{sqrt{2pi}}e^{-x^2/2}frac1{sqrt{2pi}}e^{-z^2/2},dz,dx.
$$
Switching to polar coordinates, this equals
$$
begin{align}
int_{theta=arctan(a)}^{pi/2}int_{r=0}^inftyfrac1{2pi}e^{-r^2/2},rdrdtheta&=int_{theta=arctan(a)}^{pi/2}frac1{2pi},dtheta\
&=frac1{2pi}left(fracpi2-arctan aright)\
&=frac14+frac1{2pi}arctanfracrho{sqrt{1-rho^2}};
end{align}
$$
for the last equality we substitute $a:=frac{-rho}{sqrt{1-rho^2}}$ and use the fact that the arctan function is odd. To finish off, remember that if $thetain[-pi/2,pi/2]$ then $$theta=arctanfrac{rho}{sqrt{1-rho^2}} Longleftrightarrow tantheta=frac{rho}{sqrt{1-rho^2}} Longleftrightarrow sintheta=rho Longleftrightarrow theta=arcsin rho.$$
$endgroup$
Using the OP's hint: The event ${X>0,Y>0}$ is the same as the event ${X>0,Z>frac{-rho}{sqrt{1-rho^2}}X}$, where now $X$ and $Z$ are independent standard normal variables. Writing $a:=frac{-rho}{sqrt{1-rho^2}}$ for brevity, the desired probability is expressible as a double integral involving the joint density of $(X,Z)$:
$$
P(X>0,Y>0)=P(X>0,Z>aX)=int_{x=0}^inftyint_{z=ax}^inftyfrac1{sqrt{2pi}}e^{-x^2/2}frac1{sqrt{2pi}}e^{-z^2/2},dz,dx.
$$
Switching to polar coordinates, this equals
$$
begin{align}
int_{theta=arctan(a)}^{pi/2}int_{r=0}^inftyfrac1{2pi}e^{-r^2/2},rdrdtheta&=int_{theta=arctan(a)}^{pi/2}frac1{2pi},dtheta\
&=frac1{2pi}left(fracpi2-arctan aright)\
&=frac14+frac1{2pi}arctanfracrho{sqrt{1-rho^2}};
end{align}
$$
for the last equality we substitute $a:=frac{-rho}{sqrt{1-rho^2}}$ and use the fact that the arctan function is odd. To finish off, remember that if $thetain[-pi/2,pi/2]$ then $$theta=arctanfrac{rho}{sqrt{1-rho^2}} Longleftrightarrow tantheta=frac{rho}{sqrt{1-rho^2}} Longleftrightarrow sintheta=rho Longleftrightarrow theta=arcsin rho.$$
answered Dec 12 '18 at 22:44
grand_chatgrand_chat
20.4k11327
20.4k11327
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f255368%2fpx0-y0-for-a-bivariate-normal-distribution-with-correlation-rho%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
As a hint for the first question, notice that $Z$ is the conditional of $Y$ on $X$.
$endgroup$
– Learner
Dec 10 '12 at 11:03
$begingroup$
it is easy to solve part. As X and Y are standard normal variables. But I am stuck at part two.
$endgroup$
– user669083
Dec 10 '12 at 11:10
$begingroup$
If it is easy for part 1, why are you asking the question?
$endgroup$
– Learner
Dec 10 '12 at 11:10
$begingroup$
Because the second part is to use first part to solve second.
$endgroup$
– user669083
Dec 10 '12 at 11:12
$begingroup$
@user669083 Learner is right. If you want an answer for part II, then you shouldnt say: "I had to show..." You need to say "using the fact that .....", I need to show : "part II"
$endgroup$
– Seyhmus Güngören
Dec 10 '12 at 12:23