Verify that the expression formula for a hypergeometric distribution gives the same result as the general equation for the expectation of any probability distribution.
To verify that the expression formula for a hypergeometric distribution gives the same result as the general equation for the expectation of any probability distribution, we need to compare the formulas for the expectation of both.
The general equation for the expectation of any probability distribution is given by:
E(X) = Σ(x * P(X=x))
Where E(X) represents the expectation of the random variable X, x represents the possible values of X, and P(X=x) represents the probability of X taking the value x.
Now let's consider the hypergeometric distribution. It is a probability distribution that describes the probability of successes in a fixed number of draws without replacement from a finite population. It is defined by three parameters: the population size (N), the number of successes in the population (K), and the number of draws (n) made without replacement.
The expression formula for the probability mass function of the hypergeometric distribution is:
P(X=x) = (KCx) * [(N-K)C(n-x)] / (NCn)
Where (mCk) represents the binomial coefficient "m choose k" and is calculated as m! / (k! * (m-k)!)
To find the expectation of a hypergeometric distribution, we substitute the values of x and P(X=x) into the general equation:
E(X) = Σ(x * P(X=x))
E(X) = Σ(x * (KCx) * [(N-K)C(n-x)] / (NCn))
The sigma (∑) sign represents summation, indicating that we need to sum up the products of x and the corresponding probabilities for all possible values of X.
By performing the summation, we can obtain the expectation of the hypergeometric distribution.
Comparing the expression formula for the hypergeometric distribution and the general equation for the expectation, we can see that they are different. However, they describe different aspects of the distribution. The expression formula gives the probability of a specific value, while the general equation calculates the average value.
Therefore, to compare the results of the expression formula for the hypergeometric distribution and the general equation for the expectation, you need to calculate and compare their respective values.