Integral of pdf and cdf normal standard distribution
up vote
1
down vote
favorite
$$ int_{-infty}^{infty}Phi(a+bx)phi(x)dx=Phileft(frac{a}{sqrt{1+b^2}}right) $$
I have a problem with showing the above result, where $phi(x)$ and $Phi(x)$ respectively are the pdf and cdf of the standard normal distribution. I've tried to prove it by calculating the pdf $phi(a+bx)$ first and combine it with the pdf $phi(x)$ but the result is complicated and doesn't yield $phileft(dfrac{a}{sqrt{1+b^2}}right) $ . I really need your help. Thanks.
integration statistics normal-distribution density-function
edited Nov 15 at 18:50
Semiclassical
11k32464
asked Nov 15 at 6:12
Kildah Namariq
83
New contributor
Kildah Namariq is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
1
down vote
favorite
$$ int_{-infty}^{infty}Phi(a+bx)phi(x)dx=Phileft(frac{a}{sqrt{1+b^2}}right) $$
I have a problem with showing the above result, where $phi(x)$ and $Phi(x)$ respectively are the pdf and cdf of the standard normal distribution. I've tried to prove it by calculating the pdf $phi(a+bx)$ first and combine it with the pdf $phi(x)$ but the result is complicated and doesn't yield $phileft(dfrac{a}{sqrt{1+b^2}}right) $ . I really need your help. Thanks.
integration statistics normal-distribution density-function
edited Nov 15 at 18:50
Semiclassical
11k32464
asked Nov 15 at 6:12
Kildah Namariq
83
New contributor
Kildah Namariq is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Is $Phi(a+bx)$ pdf or cdf?
– callculus
Nov 15 at 17:11
$Phi (a+bx) $ is cdf of normal standard and $phi(x) $ is pdf of normal standard
– Kildah Namariq
Nov 15 at 17:39
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
$$ int_{-infty}^{infty}Phi(a+bx)phi(x)dx=Phileft(frac{a}{sqrt{1+b^2}}right) $$
I have a problem with showing the above result, where $phi(x)$ and $Phi(x)$ respectively are the pdf and cdf of the standard normal distribution. I've tried to prove it by calculating the pdf $phi(a+bx)$ first and combine it with the pdf $phi(x)$ but the result is complicated and doesn't yield $phileft(dfrac{a}{sqrt{1+b^2}}right) $ . I really need your help. Thanks.
integration statistics normal-distribution density-function
edited Nov 15 at 18:50
Semiclassical
11k32464
asked Nov 15 at 6:12
Kildah Namariq
83
New contributor
Kildah Namariq is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$$ int_{-infty}^{infty}Phi(a+bx)phi(x)dx=Phileft(frac{a}{sqrt{1+b^2}}right) $$
I have a problem with showing the above result, where $phi(x)$ and $Phi(x)$ respectively are the pdf and cdf of the standard normal distribution. I've tried to prove it by calculating the pdf $phi(a+bx)$ first and combine it with the pdf $phi(x)$ but the result is complicated and doesn't yield $phileft(dfrac{a}{sqrt{1+b^2}}right) $ . I really need your help. Thanks.
integration statistics normal-distribution density-function
integration statistics normal-distribution density-function
edited Nov 15 at 18:50
Semiclassical
11k32464
asked Nov 15 at 6:12
Kildah Namariq
83
New contributor
Kildah Namariq is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Nov 15 at 18:50
Semiclassical
11k32464
asked Nov 15 at 6:12
Kildah Namariq
83
New contributor
Kildah Namariq is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Nov 15 at 18:50
Semiclassical
11k32464
edited Nov 15 at 18:50
Semiclassical
11k32464
edited Nov 15 at 18:50
Semiclassical
11k32464
11k32464
asked Nov 15 at 6:12
Kildah Namariq
83
New contributor
Kildah Namariq is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 15 at 6:12
Kildah Namariq
83
asked Nov 15 at 6:12
Kildah Namariq
83
83
New contributor
Kildah Namariq is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Kildah Namariq is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Kildah Namariq is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Is $Phi(a+bx)$ pdf or cdf?
– callculus
Nov 15 at 17:11
$Phi (a+bx) $ is cdf of normal standard and $phi(x) $ is pdf of normal standard
– Kildah Namariq
Nov 15 at 17:39
add a comment |
Is $Phi(a+bx)$ pdf or cdf?
– callculus
Nov 15 at 17:11
$Phi (a+bx) $ is cdf of normal standard and $phi(x) $ is pdf of normal standard
– Kildah Namariq
Nov 15 at 17:39
Is $Phi(a+bx)$ pdf or cdf?
– callculus
Nov 15 at 17:11
Is $Phi(a+bx)$ pdf or cdf?
– callculus
Nov 15 at 17:11
$Phi (a+bx) $ is cdf of normal standard and $phi(x) $ is pdf of normal standard
– Kildah Namariq
Nov 15 at 17:39
$Phi (a+bx) $ is cdf of normal standard and $phi(x) $ is pdf of normal standard
– Kildah Namariq
Nov 15 at 17:39
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
Using $N(0,,1)$ iids $U,,V$ and defining $W:=U-bVsim N(0,,1+b^2)$, we can write the integral as $$int_{Bbb R}P(Ule a+bx)dP(Vle x)=P(Ule a+bV)=P(Wle a)=Phibigg(frac{a}{sqrt{1+b^2}}bigg).$$
answered Nov 15 at 19:40
J.G.
18k11830
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Using $N(0,,1)$ iids $U,,V$ and defining $W:=U-bVsim N(0,,1+b^2)$, we can write the integral as $$int_{Bbb R}P(Ule a+bx)dP(Vle x)=P(Ule a+bV)=P(Wle a)=Phibigg(frac{a}{sqrt{1+b^2}}bigg).$$
answered Nov 15 at 19:40
J.G.
18k11830
add a comment |
up vote
2
down vote
accepted
Using $N(0,,1)$ iids $U,,V$ and defining $W:=U-bVsim N(0,,1+b^2)$, we can write the integral as $$int_{Bbb R}P(Ule a+bx)dP(Vle x)=P(Ule a+bV)=P(Wle a)=Phibigg(frac{a}{sqrt{1+b^2}}bigg).$$
answered Nov 15 at 19:40
J.G.
18k11830
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Using $N(0,,1)$ iids $U,,V$ and defining $W:=U-bVsim N(0,,1+b^2)$, we can write the integral as $$int_{Bbb R}P(Ule a+bx)dP(Vle x)=P(Ule a+bV)=P(Wle a)=Phibigg(frac{a}{sqrt{1+b^2}}bigg).$$
answered Nov 15 at 19:40
J.G.
18k11830
Using $N(0,,1)$ iids $U,,V$ and defining $W:=U-bVsim N(0,,1+b^2)$, we can write the integral as $$int_{Bbb R}P(Ule a+bx)dP(Vle x)=P(Ule a+bV)=P(Wle a)=Phibigg(frac{a}{sqrt{1+b^2}}bigg).$$
answered Nov 15 at 19:40
J.G.
18k11830
answered Nov 15 at 19:40
J.G.
18k11830
answered Nov 15 at 19:40
J.G.
18k11830
answered Nov 15 at 19:40
J.G.
18k11830
18k11830
add a comment |
add a comment |
Kildah Namariq is a new contributor. Be nice, and check out our Code of Conduct.
Kildah Namariq is a new contributor. Be nice, and check out our Code of Conduct.
Kildah Namariq is a new contributor. Be nice, and check out our Code of Conduct.
Kildah Namariq is a new contributor. Be nice, and check out our Code of Conduct.
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%2f2999304%2fintegral-of-pdf-and-cdf-normal-standard-distribution%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
Is $Phi(a+bx)$ pdf or cdf?
– callculus
Nov 15 at 17:11
$Phi (a+bx) $ is cdf of normal standard and $phi(x) $ is pdf of normal standard
– Kildah Namariq
Nov 15 at 17:39