The continuous type variable can take any value in a certain interval, so its probability distribution cannot be expressed by the distribution column, only by the probability distribution density curve.

**1. Normal distribution**

The normal distribution is one of the most common and most important continuous distributions, and the probability density function is as follows:

The cumulative probability distribution function is as follows:

The normal distribution has two parameters, μ and σ. We can represent the normal distribution as n (μ,σ). When μ=0,σ=1, such a normal distribution is called the standard normal distribution

**2. Exponential distribution**

Exponential distribution is used to denote the time interval of an independent random event, and its density function decreases exponentially as the value becomes larger.

Where λ> 0 is a parameter of the distribution, often referred to as the rate parameter (rates parameter). That is, the number of times an event occurs per unit of time. The interval of the exponential distribution is [0,∞]. If a random variable X is exponentially distributed, you can write: the x~ exponential (λ) cumulative probability distribution function is as follows:

The exponential distribution is a special case of gamma distribution and weibull distribution, and the product's failure is accidental failure, and its lifetime is subject to exponential distribution. The exponential distribution can also be considered as a special distribution of the shape coefficients equal to 1 in the Weibull distribution, the failure rate of the exponential distribution is a constant independent of the time t, so the distribution function is simple.

An important feature of exponential functions is the memory-free (also known as loss of memory). This means that if a random variable is exponentially distributed, there is P (t>s+t| when s,t≥0) t>t) =p (t>s), i.e., if T is the life of a component, it is known that the component uses a T-hour, and that it uses a conditional probability of at least s+t hours, equal to the probability that it uses at least s-hours from the time it was used. Obviously, this characteristic of exponential distribution is completely contradictory to the actual condition of fatigue, wear, corrosion and creep of mechanical parts, which violates the process of product damage accumulation and aging. Therefore, the exponential distribution can not be used as a functional parameter distribution of mechanical parts, but it may be approximated as a high reliability of complex components, machine or system failure distribution model, especially in parts or machines in the whole machine test is widely used.

**3.Gamma (gamma) distribution**

Gamma distribution (gamma) is an important member of the famous Pearson probability distribution function cluster, known as the Pearson Ⅲ type distribution. Its curve has a peak, but the left and right asymmetry. The parameter α in the gamma distribution is called the shape parameter, and β is called the scale parameter.

The Gamma function is:

The probability density function is as follows:

The cumulative distribution function is as follows

Gamma distribution has the characteristics of exponential distribution and power distribution. From the gamma distribution formula, when Beta is zero, it becomes a power distribution. When N=1, it becomes an exponential distribution. and its distribution function is the product of the first two distributions (coefficients are adjusted). Power Distribution and exponential distribution in variable values are very small, but the maximum value of the gamma they make is not the smallest variable, but a peak is centered.

**4. Uniform distribution**

Uniform distribution is a simple probability distribution of uniformity, non-deviation, divided into discrete uniform distribution and continuous uniform distribution, the main characteristics are: The measured value in a certain range of opportunities appear everywhere, that is, uniform, it is also known as rectangular distribution or equal probability distribution.

Probability density function

Cumulative distribution function

**5.weibull distribution**

Weibull (Weber) distribution, also known as Merriam-Webster distribution or Weibull distribution, is the theoretical basis for reliability analysis and life test. The Weibull distribution can be applied to many forms, the distribution is determined by the shape, scale (range) and position of three parameters. The shape parameter k is the most important parameter, determines the basic shape of the distribution density curve, the scale parameter λ enlarges or reduces the function of the curve, but does not affect the shape of the distribution.

Weibull distributions are typically used in the Fault Analysis field (field of failure analyses), and in particular it can simulate (mimic) the distribution of a continuous (over time) variation of the failure rate (failture rates).

The failure rate is:

1. Constant (constant over time), then α=1, implies that a random event occurs

2. has been reduced (decreases over time), then α<1, implying "early failure (infant mortality)"

3. has been increased (increases over time), then α>1, implying "exhaustion (wear out)"-the likelihood of failure becomes larger with the advance

Probability density function

Cumulative distribution function

Weibull distribution is related to many distributions. For example, when k=1, it is exponential distribution; k=2 is Rayleigh distribution (Rayleigh distribution). X is a random variable, which is a positional parameter, which can be positively negative, usually positive or equal to zero, and a positive value represents a time delay, short time lag.

**6.β (Beta beta) distribution**

Beta distribution refers to a set of continuous probability distributions defined in the (0,1) interval, with beta distributions of α and β two parameter α,β>0, where α is the number of successes plus 1,β for failures plus 1.

An important part of the beta distribution should be the presence of conjugate prior distributions as Bernoulli distributions and two-term distributions, which have important applications in machine learning and mathematical statistics. The parameters in the beta distribution can be interpreted as pseudo-counts, and the likelihood function of the Bernoulli distribution can be expressed as the probability of an event occurring, which has the same form as the beta, so the beta distribution can be used as its prior distribution.

Probability density function

The random variable x obeys the parameter a,β, obeys the beta distribution, gamma is the gamma function

The cumulative distribution function is as follows

**7.F distribution**

The F-Distribution is presented by Fisher and is widely used in likelihood ratio testing, especially in ANOVA. The F distribution is defined as: set X, y as two independent random variables, x obey the chi-square distribution of degrees of freedom K1, y obey the K2 distribution of degrees of freedom, these 2 independent chi-square distributions are separated by their degrees of freedom in addition to the ratio of this statistic distribution. That is: the upper type F obeys the first degree of freedom is K1, the second degree of freedom is K2 F distribution.

Properties of the F-distribution

1. It is an asymmetric distribution

2. It has two degrees of freedom, namely N1-1 and n2-1, the corresponding distributions are recorded as F (n1–1, n2-1), n1–1 is often referred to as molecular degrees of freedom, and n2-1 is often referred to as the denominator degree of freedom

The 3.F distribution is a distribution family with degrees of freedom n1–1 and n2-1, and different degrees of freedom determine the shape of the F-distribution.

The reciprocal nature of the 4.F distribution:

Probability density function

B is beta functions (beta function)

Cumulative distribution function

**8.T distribution**

Student T-distribution (Student ' s t-distribution), which can be referred to as T distribution. The average number of parent groups that are estimated to be normally distributed. It is the basis of a student's T-test for a significant difference of two sample mean values. The student's T-Test improved the Z-calibration (z-test) because the Z-calibration was known as the precondition of the mother's standard deviation. Although the number of samples is large (more than 30), the z-calibration can be used to obtain the approximate value, but the Z-verification used in the small sample will produce a great error, so the student T-Test must be replaced with accuracy. In cases where the standard deviation of the mother is unknown, the student T-test can be applied regardless of the number of samples or small size. When the data to be compared has more than three groups, because the error cannot be depressed, at this time can use the Mutation number analysis (ANOVA) instead of the student T-Test.

Probability density function

V equals n-1. The distribution of T is called the T-distribution. Parameter \nu are generally referred to as degrees of freedom.

Gamma is the gamma function.

Cumulative distribution function

V equals n-1. The distribution of T is called the T-distribution. Parameter \nu are generally referred to as degrees of freedom.

Gamma is the gamma function.

Characteristics of t distribution

1. With 0 as the center, the symmetrical single-peak distribution of the left and right;

The shape of the 2.t distribution curve is related to the size of N (exactly, the degree of Freedom V). Compared with the standard normal distribution curve, the lower the Freedom V, the flatter the T distribution curve, the lower the middle of the curve, the higher the end of the curve, the more the T distribution curve is closer to the normal distribution curve, and the T distribution curve is the normal normal distribution curve when the degree of freedom v=∞.

**9.χ² (Chi-square) distribution**

If n independent random variables ξ 1, ξ2 、......、 ξn, are subject to the standard normal distribution (also known as independent distribution in the standard normal distribution), then this n obeys the standard normal distribution of the sum of squares of random variables and constitute a new random variable, the distribution of the law is called χ² distribution (chi-square Distribution). where parameter n is called degrees of freedom, the difference of degrees of freedom is another χ² distribution, just as the average or variance in a normal distribution is another normal distribution.

Chi-square distribution is a new distribution constructed from the normal distribution, when the degree of Freedom N is large, the distribution is approximately normal.

Probability density function

Gamma is the gamma function.

Cumulative distribution function

Gamma is the gamma function.

Continuous type variable distribution