Thread: Real valued Random Varaible - Please chech my definition

I am trying to understand Random Variables.

Please do let me know if my definiton is correct - or I have got things mixed up.

1. Let be the sample space i.e. set of all possible outcomes of an experiment; This can be countable or uncountable.

2. be the sample point i.e. a specific outcome of the experiment

A real valued random variable is a
Function,

And that is it !!

Informally for every , there is a unique

Is this definition correct / exact ?

Reason of my confusion is that some books define random variable as "a funciton which maps event in the given sample space to a real number." - I think this definition is wrong. Any comments?

I am trying to understand two concepts
1. E(X) - Expected value of X
2. E(X|A) - Conditional expectaion under event A

I think in 2 above, we have not changed the way X was defined on the sample space, only thing that changed is probabiltiy measure - Is this understanding correct?

Please help - Any references would be welcome

Hello,

It's ok for the definition of .

For the definition of a real valued random variable, it's correct for the first part : it's a function from to . But I don't agree with the second part : for , there is a unique . It's just a function.

If you want to go further, you'll have to talk about measurability of the random variable. I don't know if you know this though.

I am trying to understand two concepts
1. E(X) - Expected value of X
2. E(X|A) - Conditional expectaion under event A

I think in 2 above, we have not changed the way X was defined on the sample space, only thing that changed is probabiltiy measure - Is this understanding correct?

No, in general, A will be another random variable, not an event. At least that's how conditional expectation works.
E(X|A) will be another random variable. You can keep the probability measure, because the sample space isn't much modified.
*In a non-formal way* The thing that changes is that in the "information" of X, we only keep the parts that are "related" to A.
E(X|A) is viewed as the orthogonal projection of X in

I guess this is a bit abstract, but since I don't know what your background is, I just hope you will find your answer among all this stuff

Oops. I didn't follow your "orthogonal projection of X" part. Guess need to do some reading. Also I always thought conditional expectation is always wrt to an event.

I have basic idea of analysis and am trying to learn these things on my own. Any good reference you can suggest plz - formal but not too technical. For e.g. I do not know measure theory - so can I still pick this stuff?

If you can please answer this questions -

Consider a probability space -

Question - so is something like E(X|A) defined? Or this is something which is not defined.

To be precise -
1. By X, I mean a real random variable.
2. By A, I mean an event i.e. an element of sigma-algebra, F

Okay, I'll try to answer more precisely.. But I'm not a teacher so I don't have all the experience of explaining this stuff

Originally Posted by aman_cc

Oops. I didn't follow your "orthogonal projection of X" part. Guess need to do some reading.

It doesn't matter then, it was sort of an extra.

Also I always thought conditional expectation is always wrt to an event.

That's for conditional probability. It is possible to define a conditional expectation wrt an event, but in advanced probability, it is "generalized" to conditional expectation wrt a random variable (more precisely, the sigma-algebra generated by the random variable)

(quote]I have basic idea of analysis and am trying to learn these things on my own. Any good reference you can suggest plz - formal but not too technical. For e.g. I do not know measure theory - so can I still pick this stuff?[/QUOTE]
Well there is a time when you have to know some basics of measure theory.
I don't know books, especially in English since English is not my native tongue. I've heard of Durrett's Probability : Theory and examples. But I can't say if it's the best.

Originally Posted by aman_cc

If you can please answer this questions -

Consider a probability space -

Okay, so you have the F in your definitions. Then I'll add some things to my first post.

What is a real valued random variable X ?
Yes it's a function, but a function from the probability space to
is a sigma-algebra related to the space , same for , a sigma-algebra related to .
X will map from to , but it is also measurable. Which means that for any , the set will belong to .
When you do exercises, you don't need this though ! But I'm giving you the complete definition of a real valued random variable.

Question - so is something like E(X|A) defined? Or this is something which is not defined.

To be precise -
1. By X, I mean a real random variable.
2. By A, I mean an event i.e. an element of sigma-algebra, F

Yes it is defined.
It is defined as .

Thanks a lot. I have followed a few things - but guess I need to do a bit more reading before I can move ahead on this subject. Thanks very much for your help !