# Math Help - Real valued Random Varaible - Please chech my definition

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

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

A real valued random variable is a
Function, $X:\Omega \rightarrow \mathbb{R}$

And that is it !!

Informally for every $\omega \in \Omega$, there is a unique $X(\omega) \in \mathbb{R}$

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?

2. Hello,

It's ok for the definition of $\Omega$.

For the definition of a real valued random variable, it's correct for the first part : it's a function from $\Omega$ to $\mathbb R$. But I don't agree with the second part : for $\omega$, there is a unique $X(\omega)$. 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 $L^2(A)$

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

3. 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?

Consider a probability space - $(\Omega, F, P)$

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

5. 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
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.

6. Originally Posted by aman_cc

Consider a probability space - $(\Omega, F, P)$
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 $(\Omega,\mathcal F,P)$ to $(\mathbb R,\mathcal E$
$\mathcal F$ is a sigma-algebra related to the space $\Omega$, same for $\mathcal E$, a sigma-algebra related to $\mathbb R$.
X will map from $\Omega$ to $\mathbb R$, but it is also measurable. Which means that for any $E\in\mathcal E$, the set $X^{-1}(E)=\{\omega\in\Omega,X(\omega)\in E\}$ will belong to $\mathcal F$.
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 $\displaystyle \frac{1}{P(A)}\cdot \int_A X dP=\frac{1}{P(A)} \cdot E[X\mathbf{1}_{X\in A}]$.

7. 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 !