**3.1 Two-Point Distribution and even distribution**

1. Two-Point Distribution

Many random events have only two results. If the result of the product is qualified or unqualified, the product or reliable work may fail. This type of random event variable has only two values. Generally, 0 and 1 are used. It follows two-point distribution.

Its probability distribution is:

Where Pk = P (X = Xk) indicates the probability that X obtains the Xk value:

0 ≤ P ≤ 1.

Expected E (X) = P of X

Variance of x d (X) = P (1-P)

2. Even Distribution

If the probability density function f (X) of the continuous random variable x is equal to a constant in a finite range [a, B], the distribution of X is even.

Its probability distribution is:

Expected E (X) = (a + B)/2 of X

Variance of x d (X) = (B-a) 2/12

**3.2 Distribution of application in sampling test**

3.2.1 hyper-Geometric Distribution

Assume that there are a batch of products with N in total and d in the number of unqualified products. n samples are randomly extracted from these products as the samples to be inspected, the number of nonconformities X in the sample follows the Hypergeometric Distribution.

The probability of X distribution is:

X = 0, 1 ,......

Expected E (X) of X = nd/N

Variance D (X) = (nd/N) (N-d)/N) (N-n)/N) (1/2) of X)

**3.2.2 binary Distribution**

The probability formula of the Hypergeometric Distribution can be written as a factorial with a total of nine factorial values. Therefore, the calculation is cumbersome. The two-item distribution can be seen as a simplification of the Hypergeometric Distribution.

Assume that there are a batch of products with a nonconforming product rate of P. n samples are randomly extracted from these products as the samples to be inspected. The number of nonconforming products X follows two distributions.

The Probability Distribution of X is:

0 <p <1

X = 0, 1 ,......, N

X's expected E (X) = np

Variance of x d (X) = np (1-p)

**3.2.3 Poisson distribution**

Poisson distribution is more important than binary distribution. We introduced Poisson distribution from the fact that the product fails due to impact (I .e. instantaneous high voltage, high ambient stress, high load stress, etc. Assume that the product fails only after a certain number of impacts, and the impact meets three conditions:

(1) The number of times of product impact is independent of each other within two non-overlapping time intervals;

(2) chances of two or more impacts within a sufficiently small interval are negligible;

(3) The average number of times of an impact within a unit of time λ (λ> 0) does not change with time, that is, the average number of times of an impact occurs within the time interval △t, it has nothing to do with the starting point of △t.

The number of times X of the impact occurred within the time range [0, t] follows the Poisson distribution, and the probability of the impact distribution is:

X's expected E (X) = λ t

Variance of x d (X) = λ t

Assuming that the instrument is affected for n times, the reliability of the instrument within the time range [0, T] is:

Where: x = 0, 1, 2 ,......, Lambda> 0, T> 0.

**3.2.4 x2 Distribution**

This distribution is one of the most common distributions in reliability engineering. Although its probability density form is complex, it can be introduced by the standard normal distribution.

V random variables x1, x2 ,...... XV, which is subject to standard normal distribution N ). Note X2 = X12 + x22 +... When xv2 and X2 are read as "chi-square", the distribution of X2 is called the X2 distribution. Its probability density function is:

This formula is called the X distribution where the random variable X2 is subject to Degrees of Freedom v.

Formula: V-is a degree of freedom, which is a natural number.

The most important feature of X2 distribution is:

When M is an integer:

**3.3 Product Life Distribution**

3.3.1 Exponential Distribution

Exponential distribution is the most important distribution of electronic products in reliability engineering. In general, the lifetime of an electronic product is subject to the exponential distribution rule in the random failure phase after the early fault is eliminated and before the component or material is aged and deteriorated.

Exponential distribution is the only probability distribution of continuous random variables whose failure efficiency does not change with time. Easy to launch:

Exponential Distribution has the following three features:

1. The mean life and failure rate are reciprocal;

MTBF = 1/λ

2. The feature life is the average life;

3. The exponential distribution has no memory. (That is, the previous work time of the product has no impact on the future work time)

**3.3.2 Boolean Distribution**

From the above description, we can see that the exponential distribution is only applicable to the bottom of the bath curve, but any product has an early fault and there is always a consumption failure period. In reliability engineering, use the Boolean distribution to describe the distribution of products throughout the life cycle.

Replace (-λ T) in the exponential distribution with (-(T/ETA) M) to obtain the Boolean distribution. Easy to get:

**3.3.3 normal distribution and logarithm normal distribution**

Normal distribution is also called normal distribution or Gaussian distribution. Its probability density function is:

Formula medium:-∞ <X <∞

The distribution function is recorded as follows:

Logarithm normal distribution means that if the logarithm LNT of life t is subject to normal distribution N (u, σ), then t is subject to the logarithm normal distribution. Its probability density function is:

Formula: T, σ is a positive number. μ and σ are called the "logarithm mean" and "logarithm standard deviation" of the logarithm normal distribution respectively ".

**3.4 distribution constructed for Statistical Inference**

3.4.1 tdistribution (student's distribution)

T-distribution is often used in range estimation, hypothesis test of normal population, and mechanical probability design. Obey T-distributed random variables and remember T. It is a function of random variable X2 (v) that follows the standard normal distribution N (0, 1) Random Variable U and the X2 distribution with the degree of freedom v.

Its probability density function f (t) is:

**3.4.2 F-distribution**

F distribution is mainly used for two general Hypothesis tests and variance analysis. The random variable F following the F distribution is two independent functions of the X2 Distribution Random Variable X2 (V1) and X2 (V2:

Formula: F can only take positive values. The probability density function of F distribution is:

In addition, there are β-distribution and so on.

The median rank is the median of beta-distribution. It is generally obtained using the following formula:

Median rank ≈ (i-0.3)/(n + 0.4)

Formula: n indicates the total number of samples.

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