Re: Expected Value

On Jun 8, 11:23 am, "David Jones" <dajx...@xxxxxxxxx> wrote:
DanCT wrote:
On Jun 8, 5:40 am, "David Jones" <dajx...@xxxxxxxxx> wrote:
It would be better to use some proper notation throughout. The above
is better written as the following, which shows the role of
dependence:

E(X) = Pr(T=1)*E(X|T=1) + Pr(T=0)*E(X|T=0) = p*E(X|T=1).

David Jones- Hide quoted text -

- Show quoted text -

Here's the thing though. If you right down the expression for X for
any given observation, it looks like:

X = I(T=1)* X|T=1 , where I(.) is an indicator

No, this is not a complete expresion for X: if you must, try

X = I(T=1)* X|T=1 + I(T=0)* X|T=0

... you must have an expression that covers all cases before you start taking expectations. Otherwise all you
get is the trivial expression
I(T=1)* X|T=1 = I(T=1)* X|T=1,
where you read the left hand side as "when T=1, X when T=1 is".



Taking the expectation gives:

E[X] = E[I(T=1)* X|T=1]

But,

E[I(T=1)* X|T=1] = E[I(T=1)] * E[X|T=1] {= P(T=1)*E[X|T=1], your
expression} only if X & T are independent.

See my point?

No. I don't see where independence comes into your argument, but it is starting from something that is wrong.
The rteduction to
E(X) = p*E(X|T=1),
arises from E(X|T=0) =0, which itself comes from (X|T=0) =0 .... which was your assumption. Dependence is
allowed for because X|T=1 and X|T=0 are treated explicitly and can be different.

David Jones- Hide quoted text -

- Show quoted text -


Let me try to a better job of explaining myself...

Do we agree that:

X = I(T=1)* X|T=1 + I(T=0)* X|T=0
= I(T=1)* X|T=1 + 0
= I(T=1)* X|T=1
?

Okay, then to answer your question about where independence comes in:
1. I need to find E[X]
2. Taking E[.] of both sides of the above gives... E[X] = E[I(T=1)* X|
T=1]
3. To obtain your suggestion, E(X) = p*E(X|T=1), from the above, you
need to use the property:
E[AB] = E[A]*E[B], which is true only when A & B are independent
Assuming independence results in your expression: E[X] = E[I(T=1)]
* E[X|T=1] = p*E[X|T=1]

In my problem, I cannot assume this independence.

.

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