Example of discontinuous conditional mean

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I'm looking for an example of distribution of $(X,Y)$ with density $f_XY$ such that $mathbbE[X|Y=y]$ is not continuous, but the marginal density $f_Y = int f_XY$ is continuous.

In Transference of properties from marginals to joint density functions, there is a very nice example, when $f_XY(x,y)=1y/2$ is discontinuous on $[-1,1]^2$, while $f_Y(y)=(1-|y|)$ is continuous. But in that example the conditional mean $$mathbbE[X|Y=y]=frac1)int_x^ydy=0 $$ is continuous.

Is there a good counterexample (or a way to show that it does not exist)?

calculus probability-distributions

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edited Jul 19 at 22:23

asked Jul 19 at 21:13

user298615

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up vote
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favorite

I'm looking for an example of distribution of $(X,Y)$ with density $f_XY$ such that $mathbbE[X|Y=y]$ is not continuous, but the marginal density $f_Y = int f_XY$ is continuous.

In Transference of properties from marginals to joint density functions, there is a very nice example, when $f_XY(x,y)=1y/2$ is discontinuous on $[-1,1]^2$, while $f_Y(y)=(1-|y|)$ is continuous. But in that example the conditional mean $$mathbbE[X|Y=y]=frac1)int_x^ydy=0 $$ is continuous.

Is there a good counterexample (or a way to show that it does not exist)?

calculus probability-distributions

share|cite|improve this question

edited Jul 19 at 22:23

asked Jul 19 at 21:13

user298615

305

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

I'm looking for an example of distribution of $(X,Y)$ with density $f_XY$ such that $mathbbE[X|Y=y]$ is not continuous, but the marginal density $f_Y = int f_XY$ is continuous.

In Transference of properties from marginals to joint density functions, there is a very nice example, when $f_XY(x,y)=1y/2$ is discontinuous on $[-1,1]^2$, while $f_Y(y)=(1-|y|)$ is continuous. But in that example the conditional mean $$mathbbE[X|Y=y]=frac1)int_x^ydy=0 $$ is continuous.

Is there a good counterexample (or a way to show that it does not exist)?

calculus probability-distributions

share|cite|improve this question

edited Jul 19 at 22:23

asked Jul 19 at 21:13

user298615

305

I'm looking for an example of distribution of $(X,Y)$ with density $f_XY$ such that $mathbbE[X|Y=y]$ is not continuous, but the marginal density $f_Y = int f_XY$ is continuous.

In Transference of properties from marginals to joint density functions, there is a very nice example, when $f_XY(x,y)=1y/2$ is discontinuous on $[-1,1]^2$, while $f_Y(y)=(1-|y|)$ is continuous. But in that example the conditional mean $$mathbbE[X|Y=y]=frac1)int_x^ydy=0 $$ is continuous.

Is there a good counterexample (or a way to show that it does not exist)?

calculus probability-distributions

share|cite|improve this question

edited Jul 19 at 22:23

asked Jul 19 at 21:13

user298615

305

share|cite|improve this question

share|cite|improve this question

edited Jul 19 at 22:23

edited Jul 19 at 22:23

edited Jul 19 at 22:23

asked Jul 19 at 21:13

user298615

305

asked Jul 19 at 21:13

user298615

305

asked Jul 19 at 21:13

user298615

305

305

add a comment | 

add a comment | 

1 Answer
1

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up vote
2
down vote



accepted

Let $Ysim U(-1,1)$ and $Xmid Y=ysim N(1y>0, 1)$. Then

$$
mathsfE[Xmid Y=y]=1y>0,
$$
$$
f_X(x)=frac12phi(x)+frac12phi(x-1), quadtextandquad f_Y(y)=1/2.
$$

share|cite|improve this answer

edited Jul 19 at 22:26

answered Jul 19 at 21:48

d.k.o.

7,709526

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
2
down vote



accepted

Let $Ysim U(-1,1)$ and $Xmid Y=ysim N(1y>0, 1)$. Then

$$
mathsfE[Xmid Y=y]=1y>0,
$$
$$
f_X(x)=frac12phi(x)+frac12phi(x-1), quadtextandquad f_Y(y)=1/2.
$$

share|cite|improve this answer

edited Jul 19 at 22:26

answered Jul 19 at 21:48

d.k.o.

7,709526

add a comment | 

up vote
2
down vote



accepted

Let $Ysim U(-1,1)$ and $Xmid Y=ysim N(1y>0, 1)$. Then

$$
mathsfE[Xmid Y=y]=1y>0,
$$
$$
f_X(x)=frac12phi(x)+frac12phi(x-1), quadtextandquad f_Y(y)=1/2.
$$

share|cite|improve this answer

edited Jul 19 at 22:26

answered Jul 19 at 21:48

d.k.o.

7,709526

add a comment | 

up vote
2
down vote



accepted

up vote
2
down vote



accepted

Let $Ysim U(-1,1)$ and $Xmid Y=ysim N(1y>0, 1)$. Then

$$
mathsfE[Xmid Y=y]=1y>0,
$$
$$
f_X(x)=frac12phi(x)+frac12phi(x-1), quadtextandquad f_Y(y)=1/2.
$$

share|cite|improve this answer

edited Jul 19 at 22:26

answered Jul 19 at 21:48

d.k.o.

7,709526

Let $Ysim U(-1,1)$ and $Xmid Y=ysim N(1y>0, 1)$. Then

$$
mathsfE[Xmid Y=y]=1y>0,
$$
$$
f_X(x)=frac12phi(x)+frac12phi(x-1), quadtextandquad f_Y(y)=1/2.
$$

share|cite|improve this answer

edited Jul 19 at 22:26

answered Jul 19 at 21:48

d.k.o.

7,709526

share|cite|improve this answer

share|cite|improve this answer

edited Jul 19 at 22:26

edited Jul 19 at 22:26

edited Jul 19 at 22:26

answered Jul 19 at 21:48

d.k.o.

7,709526

answered Jul 19 at 21:48

d.k.o.

7,709526

answered Jul 19 at 21:48

d.k.o.

7,709526

7,709526

add a comment | 

add a comment | 

 

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