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

  1. #1
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    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?

    Please help - Any references would be welcome
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    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
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    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?
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    If you can please answer this questions -

    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
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    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
    Quote Originally Posted by aman_cc View Post
    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.
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    Quote Originally Posted by aman_cc View Post
    If you can please answer this questions -

    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}].
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    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 !
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