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This is a very basic question in probability, but I would like a rigorous answer nevertheless.

When a text describes a random experiment, say,

  • "With probability $p$ we assign $0$ to $x_1$, and otherwise we assign $1$, and independently with probability $r$ we assign $0$ to $x_2$ and otherwise we assign $1$."

Question: How do we then define the probability distribution in terms of sample space $\Omega$, and the axioms of probability. In particular, how do we know that $$\sum_{e\in\Omega}\mathbf{Pr}\left[ e\right] =1 ?$$ Especially, when we are in the general case of $n$ such independent experiments, each chosen with probability $p_i$, $i\in[n]$. I understand that this can be computed and evaluated to 1, but is this the only "proof" of this fact? And how is this distribution called?

Jack
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  • Question is confusing. First example $P(x_1=0)=p$ then automatically $(x_1=1)=1-p$. – herb steinberg Jan 22 '22 at 00:54
  • Not sure I understand. There are only four events, right? $00, 01, 10, 11$. Just work out the probabilities of each and add. I guess we are meant to assume independence, though you didn't specify that. – lulu Jan 22 '22 at 00:57
  • @lulu, correct. Added independence. But the question is just an example of the general scenario. What if I have the general case: $n$ such experiments each with $p_i$ probability, chosen independently. Is this direct computation of the sum of the probabilities the only proof? How is this distribution/experiment called? – Jack Jan 22 '22 at 01:01
  • Prove it inductively using the same calculation. I doubt it is the only proof, but it's clear and straight forward. – lulu Jan 22 '22 at 01:02
  • @lulu, is this the established textbook proof? What's the name of this distribution/experiment? – Jack Jan 22 '22 at 01:04
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    I think this is too basic to be called an established textbook proof. I suppose I'd call it the product distribution or joint distribution (so as not to confuse it with the distribution on the product of two variables). – lulu Jan 22 '22 at 01:05
  • @lulu, yes, probably the product distribution. How do I search for the properties of this product distribution? (Actually, I think this is probably called a Bernouli trial..?) – Jack Jan 22 '22 at 01:07
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    I think you'll have better luck searching for the Joint Probability Distribution. If you search for the product, I expect you'll mostly get information on the product of two random variables. – lulu Jan 22 '22 at 01:08
  • @lulu, correct. That's what wiki gives. – Jack Jan 22 '22 at 01:09

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