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So if $X : \Omega \rightarrow T$ is a random variable mapping some sample space $\Omega$ (assume discrete and which has a probability measure defined on it) to some targets space $T$. Then the one can define its "probability mass function" as the map $p_X : T \rightarrow \mathbb{R}$ which maps $p_X(t) = P(\{ X^{-1}(t) \})$ i.e PMF of $X$ at $t$ is the probability of the event that $X$ maps to $t$.

  • Is the above correct?

  • Now let me look at a special case where I am getting confused : Say we have 1000 otherwise identical balls of which say some 200 are marked. So the probability for a random ball to be marked is $= 200/1000 = 1/5$ Then say we have a random variable X which samples some 20 of these balls at random and counts the number of marked balls.

    Now there are two natural quantities that I can think of,

    1. the probability of getting $k$ marked balls when a sample of 20 balls is drawn from the set and that is = $\frac{^{200}C_k ^{800}C_{20-k} }{ ^{1000}C_{20}} = \frac {(\# \text{ ways of creating a set of 20 balls k of which are marked})}{(\# \text{ways of creating a set of 20 balls })}$

    2. the quantity $^{20}C_k (1/5)^k(4/5)^{20-k}$ which I am not sure but probably counts the probability for a given 20 ball set to have $k$ marked balls.

Which of the above would be called the $p_X(k)$?


Is there any meaning to the quantity, $^{1000}C_{20} (1/5)^k(4/5)^{20-k}$ ?

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The explanation you give in the first paragraph is correct.

Regarding your example, 1) is the probability mass function corresponding to the described scenario. Notice how the use of $p_x$ allows us to make inferences about the scenario without knowledge of the original sample space, $\Omega$. That means that even if you formalize the experiment in other equivalent way, the probability mass function you end up with is the same!

The probability distribution you describe in 2) corresponds to what is called in probability a binomial distribution of $n$ repetititions and parameter $\theta$. It describes the outcome of situations where you sample $n$ times something, and the something has a property with probability $\theta$, and otherwise lacks the property.

For example, it would be the probability distribution of your scenario if we allowed replacement - instead of drawing 20 balls at once, draw one, note down the result, return the ball to the urn and repeat 20 times.

Regarding your last question, there is nothing immediatly related to this scenarios that comes to mind regarding that formula.

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