How to rigorously prove $E(X)=E(X|A)P(A)+E(X|A^c)P(A^c)$?
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X is a random variable while A is an event. I try to prove $$E(X)=E(X|A)P(A)+E(X|A^c)P(A^c)$$
Assuming $X$ is a continuous random variable, if $$E(X|A)=int_{xin A}xf_{X|A}(x)dx=int_{xin A}xf_{X,A}(x)/P(A)dx $$ $$E(X|A^c)=int_{xin A^c}xf_{X|A^c}(x)dx=int_{xin A^c}xf_{X,A^c}(x)/P(A^c)dx $$, and $f_{X,A}(x)=f_{X,A^c}(x)$ hold, then equality is obviously true. But the problem is I don't know whether the joint probability density function $f_{X,A}(x)$ is well defined here and if $f_{X,A}(x)=f_{X,A^c}(x)$ .
I've seen this conditional expectation equality somewhere and intuitively it makes good sense. But I hope to see a rigorous proof for it. Do I have to dig into measure theory to prove this? Thanks!
probability probability-theory conditional-expectation
edited Nov 18 at 0:47
Bernard
116k637108
asked Nov 18 at 0:46
Hank
373112
add a comment |
up vote
2
down vote
favorite
X is a random variable while A is an event. I try to prove $$E(X)=E(X|A)P(A)+E(X|A^c)P(A^c)$$
Assuming $X$ is a continuous random variable, if $$E(X|A)=int_{xin A}xf_{X|A}(x)dx=int_{xin A}xf_{X,A}(x)/P(A)dx $$ $$E(X|A^c)=int_{xin A^c}xf_{X|A^c}(x)dx=int_{xin A^c}xf_{X,A^c}(x)/P(A^c)dx $$, and $f_{X,A}(x)=f_{X,A^c}(x)$ hold, then equality is obviously true. But the problem is I don't know whether the joint probability density function $f_{X,A}(x)$ is well defined here and if $f_{X,A}(x)=f_{X,A^c}(x)$ .
I've seen this conditional expectation equality somewhere and intuitively it makes good sense. But I hope to see a rigorous proof for it. Do I have to dig into measure theory to prove this? Thanks!
probability probability-theory conditional-expectation
edited Nov 18 at 0:47
Bernard
116k637108
asked Nov 18 at 0:46
Hank
373112
For avoiding mixing random variables and events, you could define $Y=I_A$ , i.e, the indicator variable of the event.
– leonbloy
Nov 18 at 1:05
See Total expectation theorem on wikipedia or search in this site.
– StubbornAtom
Nov 18 at 5:51
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
X is a random variable while A is an event. I try to prove $$E(X)=E(X|A)P(A)+E(X|A^c)P(A^c)$$
Assuming $X$ is a continuous random variable, if $$E(X|A)=int_{xin A}xf_{X|A}(x)dx=int_{xin A}xf_{X,A}(x)/P(A)dx $$ $$E(X|A^c)=int_{xin A^c}xf_{X|A^c}(x)dx=int_{xin A^c}xf_{X,A^c}(x)/P(A^c)dx $$, and $f_{X,A}(x)=f_{X,A^c}(x)$ hold, then equality is obviously true. But the problem is I don't know whether the joint probability density function $f_{X,A}(x)$ is well defined here and if $f_{X,A}(x)=f_{X,A^c}(x)$ .
I've seen this conditional expectation equality somewhere and intuitively it makes good sense. But I hope to see a rigorous proof for it. Do I have to dig into measure theory to prove this? Thanks!
probability probability-theory conditional-expectation
edited Nov 18 at 0:47
Bernard
116k637108
asked Nov 18 at 0:46
Hank
373112
X is a random variable while A is an event. I try to prove $$E(X)=E(X|A)P(A)+E(X|A^c)P(A^c)$$
Assuming $X$ is a continuous random variable, if $$E(X|A)=int_{xin A}xf_{X|A}(x)dx=int_{xin A}xf_{X,A}(x)/P(A)dx $$ $$E(X|A^c)=int_{xin A^c}xf_{X|A^c}(x)dx=int_{xin A^c}xf_{X,A^c}(x)/P(A^c)dx $$, and $f_{X,A}(x)=f_{X,A^c}(x)$ hold, then equality is obviously true. But the problem is I don't know whether the joint probability density function $f_{X,A}(x)$ is well defined here and if $f_{X,A}(x)=f_{X,A^c}(x)$ .
I've seen this conditional expectation equality somewhere and intuitively it makes good sense. But I hope to see a rigorous proof for it. Do I have to dig into measure theory to prove this? Thanks!
probability probability-theory conditional-expectation
probability probability-theory conditional-expectation
edited Nov 18 at 0:47
Bernard
116k637108
asked Nov 18 at 0:46
Hank
373112
edited Nov 18 at 0:47
Bernard
116k637108
asked Nov 18 at 0:46
Hank
373112
edited Nov 18 at 0:47
Bernard
116k637108
edited Nov 18 at 0:47
Bernard
116k637108
edited Nov 18 at 0:47
Bernard
116k637108
116k637108
asked Nov 18 at 0:46
Hank
373112
asked Nov 18 at 0:46
Hank
373112
asked Nov 18 at 0:46
Hank
373112
373112
For avoiding mixing random variables and events, you could define $Y=I_A$ , i.e, the indicator variable of the event.
– leonbloy
Nov 18 at 1:05
See Total expectation theorem on wikipedia or search in this site.
– StubbornAtom
Nov 18 at 5:51
add a comment |
For avoiding mixing random variables and events, you could define $Y=I_A$ , i.e, the indicator variable of the event.
– leonbloy
Nov 18 at 1:05
See Total expectation theorem on wikipedia or search in this site.
– StubbornAtom
Nov 18 at 5:51
For avoiding mixing random variables and events, you could define $Y=I_A$ , i.e, the indicator variable of the event.
– leonbloy
Nov 18 at 1:05
For avoiding mixing random variables and events, you could define $Y=I_A$ , i.e, the indicator variable of the event.
– leonbloy
Nov 18 at 1:05
See Total expectation theorem on wikipedia or search in this site.
– StubbornAtom
Nov 18 at 5:51
See Total expectation theorem on wikipedia or search in this site.
– StubbornAtom
Nov 18 at 5:51
add a comment |
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For avoiding mixing random variables and events, you could define $Y=I_A$ , i.e, the indicator variable of the event.
– leonbloy
Nov 18 at 1:05
See Total expectation theorem on wikipedia or search in this site.
– StubbornAtom
Nov 18 at 5:51