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Considering the following contingency table:

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We calculate the conditional distribution for the city Manchester:

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  1. Why do we need the conditional distribution and how we interpret the result?
  2. Is conditional distribution different from conditional probability distribution?
Infinity
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1 Answers1

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  1. Why do we need the conditional distribution and how we interpret the result?

In your data, you can calculate the probability for the Manchester subgroup of belonging to the group "Football Fan" as 30/(30+50)=0.37, while the probability for other cities may differ, for example it is 24/(20+24)=0.54 for London, or 92/(92+108)=0.46 for all the data. Those are empirical probabilities, the estimates of the probabilities from your sample.

  1. Is conditional distribution different from conditional probability distribution?

In probability theory, we are dealing with probability distributions. If you are referring to the empirical (observed) distribution of the data, it can be presented as counts $n_i$ (frequency distribution) or empirical probabilities $n_i \big/ \sum_i n_i$ (probability distribution).

Tim
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  • In probability theory there are also *frequency* distributions. Although mathematically they are the same as probability distributions, they are applied in different ways. – whuber Dec 28 '21 at 13:55
  • Your post is ambiguous, though: if "sample from Manchester" is meant literally, your characterization is incorrect, because we don't actually know the proportion of fans in that city; if you mean "resample of this sample from Manchester," then your characterization looks correct. – whuber Dec 28 '21 at 14:36
  • It is imperative first to be *correct* before being concerned about *simplicity.* Your language risks confounding probability with frequency and confounding a sample with the population in the mind of a reader new to statistics. – whuber Dec 28 '21 at 15:30
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    @whuber I don't want to and don't think it is necessary to make the answer an introductory chapter of a probability and statistics handbook. OP asks about empirical probabilities and the answer refers to them specifically. – Tim Dec 28 '21 at 15:59
  • I am not suggesting that: certainly simplicity and accuracy can go hand in hand. Being correct does not require being any more verbose. I fear that introducing the phrase "empirical probabilities" only complicates the explanation rather than making it simpler or clearer. Indeed, why even introduce the concept of estimation in a setting where no estimates are called for? – whuber Dec 28 '21 at 16:13
  • @whuber your comments sound contradictory. First you suggested drawing a clear boundary between the sample vs population etc, now you suggest that it unnecessary complicates the answer. – Tim Dec 28 '21 at 16:22
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    I'm sorry that I'm not communicating well in this dialog -- but I do think you are reading things into my comments that aren't there. I have been arguing for the merits of *simplicity,* *clarity,* and *correctness* -- that's all. – whuber Dec 28 '21 at 16:24