What Is the Geometric Distribution?

Geometric distribution is a discrete probability distribution. One of them is defined as: in n Bernoulli trials, k trials yield the probability of first success. In detail, it is: the probability of failure in the first k-1 times and success in the k-th time. The geometric distribution is a special case of Pascal distribution when r = 1.

In the Bernoulli test, note that the probability of event A occurring in each test is p, and the test stops when event A occurs. The number of tests performed at this time is X, and the distribution is

This distribution sequence is a general term of a geometric sequence, so X is said to obey a geometric distribution, and it is denoted as X ~ GE (p).

In practice, many random variables obey the geometric distribution. For example, if the failure rate of a certain product is 0.05, the number of inspections for the first time that a defective product is found X GE (0.05).

It has two cases:

(1) N times to get 1 success

For event A with probability p, take X as the number of trials for the first occurrence of A, then the distribution of X:

,

With this distribution

1 Generalization of geometric distribution 1

The following test is performed. In the Bernoulli test, the probability of occurrence of event A in each test is recorded as p, and the test proceeds to events A and

Stop when all appear, and the number of tests performed at this time is X, then:

Among them, q = 1-p, k = 2, 3, ...

Therefore, the above formula can be a distribution sequence. This distribution sequence is the sum of two general terms of the geometric sequence. Here we call X obey the generalized geometric distribution under two events. Let X = PGE (2; p). The mathematical expectation is :

. When P =

When E (X) takes the minimum value, E (X) = 3.

due to

, So you get:

2 Generalization of geometric distribution 2

An independent repetitive test is now performed, and one of three events A, B, and C will occur in each test. Record the probability of events A, B, and C in each test as

,

And

. The test stops after events A, B, and C all occur. At this time, the number of tests performed is X, and there are:

Where k = 3,4, ... Therefore, the above formula can also be used as a distribution sequence. This distribution sequence is the sum and difference of the general terms of six geometric series. It is said that X obeys the generalized geometric distribution under three events, and is denoted as X PGE (3;

). The mathematical expectations are:

Easy to verify when

When E (X) has a minimum value, E (X) = 5.5 ^[1] .