Bernoulli test, binomial distribution

Bernoulli experiment

If experiment E has only two possible outcomes: (occurs) and (does not occur, then reverse event), then E is called Bernoulli trial.

Suppose occurs with probability , where , then The probability of occurrence is.

Repeating experiment E n times independently is called n-fold Bernoulli trial.

Repetition: means that in each trial, the probability remains the same. For example, if you flip a coin, the probability of it coming up remains the same every time. With replacement sampling, the probability of drawing a red ball each time remains unchanged. But if sampling is done without replacement, then the probability of drawing a red ball changes each time.

Independent: means that the results of each test do not affect each other. For example, when flipping a coin, the results do not affect each other. With replacement sampling, the results each time have no influence on each other. But if it is sampling without replacement, then the results each time will affect each other.

In n-fold Bernoulli trials, assume that the result of the th trial is , where Equal to or , independence means:

binomial distribution

In an n-fold Bernoulli trial, assuming that the random variable represents the number of times the event occurs, then its probability is:

,in

The random variable obeys the binomial distribution with parameters n and p, and is recorded as .

Remarks:The reason why the above formula is called the binomial distribution is because is exactly equal to the binomialThe term that contains in the expansion (replace 1-p with q).

When n=1, it means that only one test was conducted, and the above formula becomes:

,in

This becomes a (0-1) distribution. Therefore, the (0-1) distribution is a special case of the binomial distribution.