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So almost everywhere in the book it's written "random variables are IID", what does this mean?

I think it means independent and identically distributed but not sure.

So by definition A and B R.V are independent means that:

$p(A\cup B)=p(A)+p(B)$ right?

But what does identically distributed mean? Does it mean that the variables have the exact same distribution?

Thanks a lot!

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    $\begingroup$ wiki:/iid rv $\endgroup$ – Sasha Oct 2 '13 at 14:34
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    $\begingroup$ "by definition A and B R.V are independent means that: p(A∪B)=p(A)+p(B) right?" No, absolutely not right. $\endgroup$ – Did Oct 2 '13 at 15:52
  • $\begingroup$ Okay, it's $p(a,b)=p(a)*p(b)$ but can you please tell me why it's not right? $\endgroup$ – wwbb90 Oct 2 '13 at 17:39
  • $\begingroup$ I mean doesn't it directly follow from the definition as an equivalent definition? $\endgroup$ – wwbb90 Oct 2 '13 at 17:39
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It means independent and identically distributed. You are correct.

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a probability being i.i.d (independent and identically distributed) can basically be expressed in two steps:

1) when the outcomes of a random variable does not affect each other "independent".

2) when the outcomes share the same distribution with the same parameters. For example, assume the distribution to be N(0,1/2), that is normal with mean=0 and variance=1.

I will give you a concrete example of (i.i.d.), think of tossing a coin 'n' number of times. Now, in this case, our random variable is "coin", the probability of having head is =1/2 and the probability of having no head is = 1/2. Therefore, they are "identically distributed"! Also, the outcomes are "independent", they do not affect each other! Therefore, this probability is (i.i.d.)!

I hope this makes it clear and easy!

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