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I have a definition of a Hypergeometric distribution as follows:

Definition: the Hypergeometric distribution is a discrete probability distribution that describes the probability of $k$ successes in $n$ draws, without replacement, from a finite population of size $N$ that contains exactly $K$ successes, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of $k$ successes in $n$ draws with replacement.

A random variable $X$: no. of successes in $K$ successes. The pdf is

$$P(X=k)=\frac{(\text{#ways for $k$ successes})\times (\text{# ways for $n-k$ failures})}{(\text{total number of way to select})}=\frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}$$

My question is what is definition of r.v $X$? In my word, I write as "A random variable $X$: no. of successes in $K$ successes. ", Is it correct? I am confusing about "no. of successes in $K$ successes" or "no. of successes in $K$ trails." Thanks in advance

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According to your book, if you have $N$ different things that can be picked, $K$ is the number of things from those $N$ things which would be considered "successes". $X$ is then the number of successes from those $K$ successes that are actually picked.

The number of trials is not $K$. The number of trials is $n$. So $X$ is the number of successes in $n$ trials, and $K$ is the number of successes "waiting to be picked" at the beginning.

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Generally, I think, this is taught as or thought of as a box full of "good" items and "bad" items. I will use a slightly different notation. Suppose we have a box full of $N$ balls, and there are $G$ good balls, with $G\leq N$. Let $X$ represent the number of "good" balls I draw when I grab $n$ balls from the box. Then if I want to calculate the probability of getting $k$ good balls in $n$ draws, then in notation it is $$P(X = k) = \frac{\binom{G}{k}\binom{N-k}{n-k}}{\binom{N}{n}}.$$ The denominator counts all the ways to choose $n$ balls from the $N$ balls in the box. The numerator counts all the ways to choose $n$ balls where there $k$ good balls and $n-k$ bad balls.


So from your perspective, if you choose one of the $K$ items, or an event from $K$ is realized, that is considered a success. You can consider it a "good" ball.

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