Discrete variable distribution

Discrete variable refers to the variable can only take discrete isolated values, usually by the number of units of measurement, such as number, number of units. Many of the distributions of discrete variables are related to last, so let's take a look at the last experiment:

In the same condition, repeated and independent experiments of each other have become last.

The key to judging whether or not the Bernoulli test is the probability of each test event A is constant, and the results of each test are unrelated to the results of the other tests, and repetition refers to the test as a series of tests, not one test, but several times, but pay attention to the probability that the recurrence of the event has no effect on each other.

1. Two distributions (binomial distribution)

The two-item distribution repeats n independent Bernoulli tests. In each trial there were only two possible outcomes, and the two outcomes were antagonistic and independent, unrelated to the results of other tests, the probability of occurrence or failure remained constant in each independent test, and this series of tests was called the N-re-knoop experiment.

The following conditions need to be met for the N-re-Knoop experiment:

1. There are only two possible outcomes of an experiment: success or failure
2. The probability of success is the same in every experiment
3. Multiple experiments are independent of each other
4. The experiment can be repeated n times
5. In n experiments, the number of successes is a random discrete variable.

Set to perform n independent tests, the probability of success per test is P (the probability of failure is 1-p). The "number of successes" in this N-test is a random variable. This random variable conforms to the two-item distribution.

The two-item distribution can be expressed as:

N-Times test, if the random variable is k, which means that the K is successful, n-k times fail. Choose K from N experiments, according to the principle of counting, there is a common possibility. Each of these may appear as a probability of

The "Two-item distribution" is named because the above P (x=k) is equal to item K of the two-item two-item expansion

Mathematical expectation of the distribution of two items e (x) =NP, Variance D (x) =NP (1-p)

is a n=20,p=0.125 distribution of two items:

2. Bernoulli distribution (Bernoulli distribution)

There are two possible outcomes. 1 indicates success, and the probability of occurrence is P (where 0<p<1). 0 indicates failure and the probability of occurrence is q=1-p. Therefore, the Bernoulli distribution can be expressed as

The Bernoulli distribution is also known as the 0-1 distribution or the two-point distribution, which is the two-item distribution at N=1, the mean E (x) =p, the variance D (x) =p (1-p)

3. Poisson distribution (Poisson distribution)

Poisson distribution is suitable for describing the number of random events in a unit time, it is the limit of two distributions, when p→0,n→+∞, while np=λ, two distributions tend to the Poisson distribution, because the Poisson distribution is a special case of two distributions, so the conditions of the two distribution are applicable to the Upauson distribution. Poisson distribution has an additive

The probability function of the Poisson distribution is:

The parameter λ is the average occurrence of random events in the unit time (or unit area)

Mathematical expectation of Poisson distribution E (x) =λ, Variance D (x) =λ

There are two characteristics of Poisson distribution:

1. The events examined are equal opportunities in any two-length interval.
2. The time spent in any one interval does not interact with each other or not, i.e., independent of each other.

The interval here is generalized, and it can represent both time and space.

Poisson distributions are used to simulate low-probability events, such as earthquakes. Earthquakes are very low probability events, we want to know a period of time, such as 10 years in the total number of earthquakes, you can divide the decade into N small time period, the probability of the occurrence of earthquakes in each time period is p. We assume that the small time period is so short that it is impossible for two earthquakes to occur in the same small time period, then the total number of earthquakes is a random variable, tending to the Poisson distribution.

"Note: only if the value of P is small and generally less than 0.1, the error generated by the Poisson distribution instead of the two distribution will be relatively small"

When p=0.075, the Poisson distribution was close to two distributions.

4. Geometric distribution (geometric distribution)

In the N-time Bernoulli test, the first chance of success was obtained by testing k times. This is the probability that the first k-1 times are unsuccessful and the K-times successful.

Let's say we conduct independent tests continuously until the test is successful. The probability of success for each test is p. So, by the time we succeed, the total number of tests performed is a random variable that can take a value of 1 to positive infinity. Such a random variable conforms to the geometric distribution

When a random variable has a value of k, it means that the previous k-1 times have failed. Therefore, we can represent the geometric distribution as:

Expectation of geometric distribution, variance

5. Negative two-item distribution (negative geometric distribution)

In the N-Times Bernoulli test, the probability of R succeeding was tested K times. The difference between a negative two-item distribution and a geometric distribution is that the geometric distribution is the total number of tests performed independently until the success occurs. A negative two-item distribution is also performed independently, but until the R-Success occurs, the total number of tests is K. When R=1, the negative two distributions are actually geometric distributions.

The expression for a negative two-item distribution is:

6. hypergeometric distribution (hypergeometric distribution)

Set Total has n, which contains m unqualified products. If you randomly do not put back to extract N products, then the number of unqualified goods k is a discrete random variable, if n≤m, then X may take 0,1,2...,n, if n>m, then may take 0,1,2...,m, by the classical method can be obtained x=k probability is:

Discrete variable distribution