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I am learning trying to learn more about statistics and probability theory, but I am having trouble understanding some of the terms that I feel have same or similar semantics just different name. For example, in picture 1. there is a contour that represents PDF function over two random variables F (body fat) and B (beer). On the right side of the picture there is marginal probability distribution for random variable F, and on the bottom of the picture there is marginal probability distribution for random variable B.

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So basically this is the way how we calculate an exact Marginal P. Distribution for entire population, correct? But because we might not know the date for entire population, we sample it. So we have an approximate curve that is show on picture 2.

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Does that mean when we say "population" we are referring to the EXACT probability distribution of the world that we wish to model mathematically? And because the data is not available, we sample the population instead and approximate it?

EDIT: Also, isn't the definition of probability distribution the following: "Probability distribution represents all possible states, and their probabilities which particular random variable can acquire". If that is true, then Probability distribution for some discrete random variable X, lets say tossing a coin, can be {HEAD, TAILS} both with probability of 0.5 of happening. What would we say a population is then in this context?

Sorry, I am just confused.

NEW EDIT: Okay so basically, I am wondering are the following statements true:

  • Sample space of some random variable X is basically a population (e.g. {heads, tails})
  • Probability distribution is defined over a sample space (i.e. population) and it represents probabilities of all possible sample values
  • In theory, we assume and analyze some probability distribution, which we believe to be true for the "world" that we are modeling
  • In practice, we sub sample the sample space (i.e. population), create a histogram, based on which we APPROXIMATE prob. distribution that we believe to be true for the entire population
  • Also, in theory we use notion of random variable to say that X can represent any possible value from sample space, while in practice (i mean statistics) notion of random variable does not exist. Instead, we refer to X= {some particular value} as a sample.
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You've basically got it!

Here's some answers to your questions:

  • No; a sample space is not a population. The sample space is the set of all the possible outcomes of an experiment. The population is the set of all events which is of interest. The set of outcomes of a coin flip {heads, tails} is a sample space, but not a population. The population would be all tosses of the coin. (Note the subtle difference: the sample space is the set of outcomes for each toss, the population is the actual results of all future tosses).

  • Correct, probability distribution is defined over a sample space and it represents probabilities of all possible sample values. But the sample space is not the population. The probability distribution tells us what the probability is of observing an outcome for each member of the population. (In your example, the probability distribution would be P(heads) = 1/2, P(tails) = 1/2, which applies to each coin flip).

  • Correct, we assume there's a probability distribution that applies to all members of the population.

  • We take a sample, which is a small group of the population, from which we compute statistics or use other methods to approximate the distribution. For example, if we didn't know that the coin were fair, we might flip the coin a few times and use the outcomes (the sample) to estimate the probability of heads.

  • Random variables are a way of representing the elements of a sample space as numbers. For example, we might say the random variable $X$ is 1 if the coin flip is heads and 0 if it is tails. But you're correct in that we use the random variables while we're doing statistics, but we don't use them in our final analysis. The whole point of random variables is to get numbers we can do math with. For example, you wouldn't say $P(X= 1) = 0.5$ to someone, because they won't know what that means. You'd say, "The probability of flipping heads is 1/2". (I hope this answers your question.)

To recap:

  • You find a coin, that may be fair or may be unfair. But you'd like to be able to say what the outcome of heads is on any given flip ("any given flip" is your population; you want to be able to apply your analysis to all flips of the coin).

  • The sample space for each coin flip is {heads, tails}. You can encode this using a random variable $X$ where $X = 1$ if the coin flip is heads and $X = 0$ if the coin flip is tails. We use the random variable because then we have numbers we can do math with.

  • Then you take a sample, by flipping the coin several times. You record the results for your random variable, which will be a sequence of 1's and 0's.

  • Now you use statistics, performed on the sequence of 1's and 0's from your sample, to estimate the probability of flipping a heads.

  • Say your estimate is 0.6. You can then say that your estimate for the probability of flipping heads each time the coin is tossed is 0.6 (or more technically, the future flips of the coin are from a Bernoulli(0.6) population). Note that this applies to all members of your population - all future tosses of the coin.

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  • $\begingroup$ Hi Nick. Thanks for the comment, please read my edit. $\endgroup$ Mar 24 '20 at 11:46
  • $\begingroup$ Hi Stefan. Check out the new response. Let me know if it answers your questions. $\endgroup$ Mar 24 '20 at 18:45
  • $\begingroup$ Nick, thank you so much for the help! I really appreciate it! I learned a lot! $\endgroup$ Mar 24 '20 at 22:42
  • $\begingroup$ So what you mean basically when you say "population is the set of all possible events", you are alluding to that population is probability distribution over all the events that are of interest (e.g. N flips of a coin). Or population does not have anything to do with probabilities at all, only outcomes of the past that we are analyzing? If that is true, then probability distribution and population are different concepts, correct? $\endgroup$ Mar 24 '20 at 22:57
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    $\begingroup$ Yes, the probability distribution and population are different concepts. The population is the group that the probability distribution applies to. And you don't find a population, you choose your population up front. Then the goal is to approximate a distribution that applies to each member of the population. I can tell you that all statisticians, and probably everyone on this site, has dealt with this confusion around the terminology when they started out. As you continue to study, these concepts will become clear in time. It just takes immersion and practice. $\endgroup$ Mar 25 '20 at 14:22

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