expectation of conditional probability

sangiobbe

New Member
Hi there,

I have two questions which I found answers to slightly different version of them in books but not to those that I have in mind so please pay attention in responding.

1) I like to calculate the expectation of conditional probability Pr(X|Y) for all values of Y. Meaning I want to integrate Pr(X|Y) over all positive Y. What is the formula for that? I found the answer to expectation of Pr(X|Y) for all values of X.
[you can consider any arbitrary probability function]

2) How can I calculate the probability of Pr(X|YZ)? Where Z and Y are two different dependent events. I found the answer to Pr(XY|Z).

Thank you

BGM

TS Contributor
Are all the X, Y, Z you mentioned in both questions are all events? Not random variable?

So what is the meaning of "for all values of Y"?

sangiobbe

New Member
No In first questions, X, Y and Z are random variables. Hence when I say all values of Y, I mean support of Y.

BGM

TS Contributor
So if X is a random variable, then X is not an event and thus

$$\Pr\{X|Y\}$$ is meaningless.

sangiobbe

New Member
It should be possible to represent events with an appropriate sigma-algebra and random variables. So if it helps consider that X, Y and Z are R.Vs.

BGM

TS Contributor
Ok anyway let $$A$$ to be an event.

Then you can compute $$g(y) \triangleq \Pr\{A|Y = y\} ~~ \forall y \in \mathcal{Y}$$
where $$\mathcal{Y}$$ is the support.

Note that this is just a function of $$y$$ define on the support. So you can replace it by the random variable and define
$$g(Y) \triangleq \Pr\{A|Y\}$$ to be another random variable on [0,1] transformed from Y. So we can always talk about the expectation:
$$E[g(Y)] = E[\Pr\{A|Y\}]$$

sangiobbe

New Member
Thanks and do you have also an answer to the second question?

BGM

TS Contributor
Which one is event? Which one is random variable?

sangiobbe

New Member
Let me try to explain it with following story from economics:
Suppose there are two states of the world High (H) and Low (L). There are two firms competing in a market charging p1 and p2 as their prices.
I like to know the probability of state H when firm one charges p1 and firm 2 charges p2. Pr(H|p1,p2)?
Therefore given my notations in the above post, X is the event, and Y and Z are random variables.

sangiobbe

New Member
BTW, are you writing symbols in your post in Latex? Do you put the syntax in between dollar sign? I am new to posting math in forums. I don't know how I can write symbols in here.

BGM

TS Contributor
In this forum those math latex code is wrapped by the pair of tags
[noparse] [/noparse]

Dason

, $$, and$$[/noparse] tags without having to add extra spaces and stuff. I think bryangoodrich was the one that introduced me to the noparse tags.