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I'm having a few conceptual difficulties with random variables and I was hoping someone could clear up a few things for me:

1) Firstly, what exactly do we mean when we say that two random variables X and Y are equal. I understand what identically distributed means, but my difficulty is with equality.

My professor says that equality of X and Y means that for every outcome ω in the sample space, X(ω) = Y(ω). Now, if these variable are continuously distributed, isn't it also true that P(X=Y) = 0 and that P( X and Y ∈ (a,b)) < 1 for (a,b) ⊂ ℝ . I don't see any inconsistency here, but it seems off. Is this really the definition of equality?

2) Also, I'm not entirely sure what it means to add two random variables. Can I go with the above and say that Z = X + Y if for every outcome ω of the sample space, Z(ω) = X(ω) + Y(ω).

3) My final conceptual difficulty is with the large sample theory. Why do we look at N observations in a population as N random variables in their own right and not as N instances of a single random variable ( which is what they intuitively seem to be ) Is this just a convenient starting point or is their a solid rationale behind it ? Surely the random variable is random and variable across the population studied, not for every individual. Does it make sense, for example, to talk about disease frequency being a random variable in the context of a single person ?

I'd appreciate any sort of clarification. Thanks :)

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# Conceptual Problems with Random Variables and Sample Theory

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