If $mathbb P{Xin Amid Y=y}$ what is $int_B mathbb P{Xin Amid Y=y}dy$?
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Let $X,Y$ two continuous r.v.
1) What is $$int_Bmathbb P{Xin Amid Y=y}dy ?$$
Is it $mathbb P{Xin Amid Yin B}$ ? It looks to be correct in discret time. What about in continuous time ?
2) Also, is $$mathbb P{Xin A, Yin B}=int_B mathbb P{Xin Amid Y=y}mathbb P{Y=y}dy ?$$
3) If $Y$ is discrete ($mathbb P{Yin D}=1$) (and $X$ continuous or discrete), does $$mathbb P{Xin A, Yin B}=sum_{yin D}mathbb P{Xin Amid Y=y}mathbb P{Y=y} ?$$
Attempts
1) $$mathbb P{Xin Amid Y=y}=int_A f_{Xmid Y}(xmid y)dx$$
and thus $$int_Bmathbb P{Xin Amid Y=y}dy=int_Aint_B f_{Xmid Y}(x,y)dxdy$$
and it should be $mathbb P{Xin Amid Yin B}$, no ?
2) and 3) I think it's by definition
probability
asked Nov 14 at 20:35
lovemath
556
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up vote
0
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favorite
Let $X,Y$ two continuous r.v.
1) What is $$int_Bmathbb P{Xin Amid Y=y}dy ?$$
Is it $mathbb P{Xin Amid Yin B}$ ? It looks to be correct in discret time. What about in continuous time ?
2) Also, is $$mathbb P{Xin A, Yin B}=int_B mathbb P{Xin Amid Y=y}mathbb P{Y=y}dy ?$$
3) If $Y$ is discrete ($mathbb P{Yin D}=1$) (and $X$ continuous or discrete), does $$mathbb P{Xin A, Yin B}=sum_{yin D}mathbb P{Xin Amid Y=y}mathbb P{Y=y} ?$$
Attempts
1) $$mathbb P{Xin Amid Y=y}=int_A f_{Xmid Y}(xmid y)dx$$
and thus $$int_Bmathbb P{Xin Amid Y=y}dy=int_Aint_B f_{Xmid Y}(x,y)dxdy$$
and it should be $mathbb P{Xin Amid Yin B}$, no ?
2) and 3) I think it's by definition
probability
asked Nov 14 at 20:35
lovemath
556
Note that with discrete r.v.'s, working with density function or cumulative function is quite the same. Unfortunately in continuous case it's quite different since $mathbb P{Y=y}=0$ (so your formula 2) is $0$). You really have to use density function ! So it would be $int_B mathbb P{Xin Amid Y=y}f_Y(y)dy$. 3) works. For 1), I don't really know what it represent, but $mathbb P{Xin Amid Yin B}=frac{mathbb P{Xin A, Yin B}}{mathbb P{Yin B}}$ that is $frac{iint_{Atimes B}f_{X,Y}(x,y)dxdy}{int_B f_Y(y)dy}$ that is not your formula...
– Surb
Nov 14 at 21:03
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $X,Y$ two continuous r.v.
1) What is $$int_Bmathbb P{Xin Amid Y=y}dy ?$$
Is it $mathbb P{Xin Amid Yin B}$ ? It looks to be correct in discret time. What about in continuous time ?
2) Also, is $$mathbb P{Xin A, Yin B}=int_B mathbb P{Xin Amid Y=y}mathbb P{Y=y}dy ?$$
3) If $Y$ is discrete ($mathbb P{Yin D}=1$) (and $X$ continuous or discrete), does $$mathbb P{Xin A, Yin B}=sum_{yin D}mathbb P{Xin Amid Y=y}mathbb P{Y=y} ?$$
Attempts
1) $$mathbb P{Xin Amid Y=y}=int_A f_{Xmid Y}(xmid y)dx$$
and thus $$int_Bmathbb P{Xin Amid Y=y}dy=int_Aint_B f_{Xmid Y}(x,y)dxdy$$
and it should be $mathbb P{Xin Amid Yin B}$, no ?
2) and 3) I think it's by definition
probability
asked Nov 14 at 20:35
lovemath
556
Let $X,Y$ two continuous r.v.
1) What is $$int_Bmathbb P{Xin Amid Y=y}dy ?$$
Is it $mathbb P{Xin Amid Yin B}$ ? It looks to be correct in discret time. What about in continuous time ?
2) Also, is $$mathbb P{Xin A, Yin B}=int_B mathbb P{Xin Amid Y=y}mathbb P{Y=y}dy ?$$
3) If $Y$ is discrete ($mathbb P{Yin D}=1$) (and $X$ continuous or discrete), does $$mathbb P{Xin A, Yin B}=sum_{yin D}mathbb P{Xin Amid Y=y}mathbb P{Y=y} ?$$
Attempts
1) $$mathbb P{Xin Amid Y=y}=int_A f_{Xmid Y}(xmid y)dx$$
and thus $$int_Bmathbb P{Xin Amid Y=y}dy=int_Aint_B f_{Xmid Y}(x,y)dxdy$$
and it should be $mathbb P{Xin Amid Yin B}$, no ?
2) and 3) I think it's by definition
probability
probability
asked Nov 14 at 20:35
lovemath
556
asked Nov 14 at 20:35
lovemath
556
edited Nov 14 at 20:54
asked Nov 14 at 20:35
lovemath
556
asked Nov 14 at 20:35
lovemath
556
asked Nov 14 at 20:35
lovemath
556
556
Note that with discrete r.v.'s, working with density function or cumulative function is quite the same. Unfortunately in continuous case it's quite different since $mathbb P{Y=y}=0$ (so your formula 2) is $0$). You really have to use density function ! So it would be $int_B mathbb P{Xin Amid Y=y}f_Y(y)dy$. 3) works. For 1), I don't really know what it represent, but $mathbb P{Xin Amid Yin B}=frac{mathbb P{Xin A, Yin B}}{mathbb P{Yin B}}$ that is $frac{iint_{Atimes B}f_{X,Y}(x,y)dxdy}{int_B f_Y(y)dy}$ that is not your formula...
– Surb
Nov 14 at 21:03
add a comment |
Note that with discrete r.v.'s, working with density function or cumulative function is quite the same. Unfortunately in continuous case it's quite different since $mathbb P{Y=y}=0$ (so your formula 2) is $0$). You really have to use density function ! So it would be $int_B mathbb P{Xin Amid Y=y}f_Y(y)dy$. 3) works. For 1), I don't really know what it represent, but $mathbb P{Xin Amid Yin B}=frac{mathbb P{Xin A, Yin B}}{mathbb P{Yin B}}$ that is $frac{iint_{Atimes B}f_{X,Y}(x,y)dxdy}{int_B f_Y(y)dy}$ that is not your formula...
– Surb
Nov 14 at 21:03
Note that with discrete r.v.'s, working with density function or cumulative function is quite the same. Unfortunately in continuous case it's quite different since $mathbb P{Y=y}=0$ (so your formula 2) is $0$). You really have to use density function ! So it would be $int_B mathbb P{Xin Amid Y=y}f_Y(y)dy$. 3) works. For 1), I don't really know what it represent, but $mathbb P{Xin Amid Yin B}=frac{mathbb P{Xin A, Yin B}}{mathbb P{Yin B}}$ that is $frac{iint_{Atimes B}f_{X,Y}(x,y)dxdy}{int_B f_Y(y)dy}$ that is not your formula...
– Surb
Nov 14 at 21:03
Note that with discrete r.v.'s, working with density function or cumulative function is quite the same. Unfortunately in continuous case it's quite different since $mathbb P{Y=y}=0$ (so your formula 2) is $0$). You really have to use density function ! So it would be $int_B mathbb P{Xin Amid Y=y}f_Y(y)dy$. 3) works. For 1), I don't really know what it represent, but $mathbb P{Xin Amid Yin B}=frac{mathbb P{Xin A, Yin B}}{mathbb P{Yin B}}$ that is $frac{iint_{Atimes B}f_{X,Y}(x,y)dxdy}{int_B f_Y(y)dy}$ that is not your formula...
– Surb
Nov 14 at 21:03
add a comment |
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Note that with discrete r.v.'s, working with density function or cumulative function is quite the same. Unfortunately in continuous case it's quite different since $mathbb P{Y=y}=0$ (so your formula 2) is $0$). You really have to use density function ! So it would be $int_B mathbb P{Xin Amid Y=y}f_Y(y)dy$. 3) works. For 1), I don't really know what it represent, but $mathbb P{Xin Amid Yin B}=frac{mathbb P{Xin A, Yin B}}{mathbb P{Yin B}}$ that is $frac{iint_{Atimes B}f_{X,Y}(x,y)dxdy}{int_B f_Y(y)dy}$ that is not your formula...
– Surb
Nov 14 at 21:03