expectation of conditional probability

[sangiobbe]

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]

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]

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]

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

\( \Pr\{X|Y\} \) is meaningless.

 

[sangiobbe]

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]

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]

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

 

[BGM]

Sorry really confused about your second question. Can you define your question clearer?
Which one is event? Which one is random variable?

 

[sangiobbe]

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]

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]

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

 

[Dason]

Another useful set of tags is the [noparse][noparse][/noparse] tag and it's corresponding ending tag. It allows us to mention the [noparse]

,

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

 

[BGM]

Yes I remember that there is a tag which allow me to put [noparse] [/noparse] in text without any parse, but just forgot the name. Thanks for the nice reminder.