The expectation of hypergeometric distribution is: \ mathbb {e} [x] = \ frac {nn _1} {n}.
I. Overview of Hypergeometric Distribution
Hypergeometric distribution is a discrete probability distribution in probability theory, which is usually used to describe random sampling and other fields. It is somewhat similar to binomial distribution, but different from binomial distribution, each experiment is not independent, but related to the previous experiment.
Second, find the expectation of hypergeometric distribution.
For a hypergeometric random variable X, which means to select n terms from the n terms owned by a * * *, and there are N_ 1 terms with certain properties, then its probability quality function is:
\ mathrm { P }(X = k)= \ dfrac { \ binom { N _ 1 } { k } \ binom { N-N _ 1 } { N-k } } { \ binom { N } { N } }
Now let's calculate its expected value.
\mathbb{e}[x]=\sum_{k=0}^{n}{k\cdot\mathrm{p}(x=k)}
Substitute the above probability quality function into:
\ begin { aligned } \ mathbb { E }[X]& amp； =\sum_{k=0}^{n}{k\cdot\frac{\binom{n_ 1}{k}\binom{n-n_ 1}{n-k}}{\binom{n}{n}}}\\&； =\sum_{k=0}^{n}{\frac{k\binom{n_ 1}{k}\binom{n-n_ 1}{n-k}}{\binom{n}{n}}}\\&；
=\sum_{k=0}^{n}{\frac{k\frac{n_ 1！ }{k！ (N_ 1-k)！ }\cdot\frac{(N-N_ 1)！ }{(n-k)！ (N-N_ 1-(n-k))！ } } { N！ }{n！ (N-n)！ } } } \ \ & amp= \ frac { nN _ 1 } { N } \ end { aligned }
Third, the proof process.
The derivation of the above formula can be accomplished by the definition of expected value and the knowledge of combinatorial mathematics.
First, let's look at the definition of expectation:
\mathbb{e}[x]=\sum_{i= 1}^{n}{x_i\mathrm{p}(x=x_i)}
Where \mathrm{P}(X=x_i) represents the probability that the random variable x takes the value of x_i. For hypergeometric distribution, the probability of each sample is not equal. For convenience, we assume that there are k projects with specific properties, so the number of schemes for selecting k projects is \binom{N_ 1}{k}.
Similarly, the number of schemes for selecting n-k unspecified projects is \binom{N-N_ 1}{n-k}. The number of schemes with n items selected from the total number of ***N items is \binom{N}{n}. Therefore, there are the following probability quality functions:
\ mathrm { P }(X = k)= \ dfrac { \ binom { N _ 1 } { k } \ binom { N-N _ 1 } { N-k } } { \ binom { N } { N } } .