1. The concept and characteristics of binomial distribution

In many experiments or observations in the field of medicine and health, people are interested in whether an event occurs:

For example, if white mice are used to test the toxicity of a certain drug, the concern is whether the white mice die or not;
A clinical trial of a new therapy to see if the patient is cured;
Observe whether the test result of an indicator is positive or not.

The occurrence of the event A we care about is called success, and the absence of it is called failure. This type of experiment is called a success-or-failure experiment . Binomial classification experiment in specified data.

2. Conditions for binomial distribution

3. The probability function of the binomial distribution

The binomial distribution means that in n independent repeated experiments that can produce only one of two possible outcomes (such as "positive" or "negative") , when the probability of "positive" remains constant for each trial , the occurrence of "positive" "The number of times X=0, 1, 2,...,, a probability distribution of n.

If a sample of size n is randomly selected from a population with a positive rate of T, then the probability distribution of the number of "positives" is X, that is, a binomial distribution, which is denoted as B(X;n, T) or B(n, T ).

4. Examples

5. Characteristics of the binomial distribution

5.1 Graphical features of the binomial distribution

n, $\pi$ are the two parameters of the binomial distribution, so the shape of the binomial distribution depends on n, $\pi$ . It can be seen that when $\pi$ = 0.5, the distribution is symmetrical and approximately symmetrical. When $\pi$ ≠0.5, the distribution is skewed, especially when n is small, $\pi$ the farther away from 0.5, the worse the symmetry of the distribution, but as long as it is not close to 1 and 0, as n increases, the distribution gradually approaches normal. Therefore, $\pi$ or 1- $\pi$ is not too small, and n is large enough, we often use the principle of normal approximation to deal with the problem of binomial distribution .