I would like to know how to use set theoretic notation to define a subset $D$ of a finite event space $S$ that contains exactly $n$ elements $d_i$ which are sampled uniformly without replacement from $S$. I am struggling to develop the appropriate notation for this, could somebody help?
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1$\begingroup$ $\{K\subseteq S| \text{card}(K) = n\}$ $\endgroup$– saulspatzFeb 13, 2019 at 17:32
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1$\begingroup$ If you are sampling without replacement, then you're just taking random $n-$element subsets. It's sampling with replacement that presents problems. $\endgroup$– saulspatzFeb 13, 2019 at 17:34
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$\begingroup$ @saulspatz how could I show that the sampling is uniform across the space, or is this implied? Sorry, I’m fairly new to this... $\endgroup$– ResquiensFeb 13, 2019 at 17:35
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1$\begingroup$ That's a different matter. Specifying the events and the specifying the probabilities of those events are always separate issues. It best just to say it in words, I think. Math is hard enough without cluttering it with unnecessary symbols. $\endgroup$– saulspatzFeb 13, 2019 at 17:39
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1$\begingroup$ No, what you have written is still sampling without replacement. For sampling with replacement, you either have to use the cartesian product, if the order in which the items are sampled is significant, or multisets, if it's not. $\endgroup$– saulspatzFeb 13, 2019 at 17:41
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1 Answer
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Fo sampling without replacement, the answer is just $$\{K\subseteq S| \text{card}(K) = n\}$$ This specifies the set of events. You also have to specify the probabilities of the events. I would simply says that all events have the same probability, or that they are equally likely.
Following up on the questions you asked it the comments, it's more complicated to describe the events for sampling with replacement. You use the cartesian product of the set with itself $n$ times, if the order in which the elements are drawn is significant, but if order doesn't matter you'd have to use multisets.
In practice, you would just say sampling with or without replacement, since anybody at all familiar with probability knows what these mean.
answered Feb 13, 2019 at 18:07