Probability distribution problems, probability problems, finding distribution rates

The probability distribution refers to the distribution of the probability of occurrence corresponding to the different outcomes of the event, which is reflected in the coordinate axis, which can intuitively see all the possible outcomes of the event and the probability of its occurrence. According to the data continuity type, the dataset can be divided into continuous type and discrete type, and the probability distribution corresponding to the outcome of the event can also be divided into continuous probability distribution and discrete regular distribution.

There are four common probability distributions: binomial distribution, Poisson distribution, geometric distribution and normal distribution.

Binomial distribution (discrete probability distribution): Suppose the experiment has only two outcomes: the probability of success is , and the probability of failure is 1-. The binomial distribution describes the probability of success x times in n trials performed independently and repeatedly.

Poisson distribution (discrete probability distribution): A Poisson distribution describes the probability that the specific number of occurrences of an event per unit time (or unit area) is k, assuming that the average number of occurrences of a known event per unit time (or area per unit area) is k.

Geometric distribution (discrete probability distribution): Suppose there is a series of Bernoulli experiments, where p is the probability of success and q=1-p is the probability of failure. The geometric distribution describes that in order to succeed in the r-th trial, the first thing to fail is r-1.

Normal distribution (continuous probability distribution): We can draw the probability distribution curve of the normal distribution, and we can see that the curve is a bell-shaped curve. If the mean, modulus, and median values of a variable are equal, then the variable is normally distributed.

1.Continuous random variables.

If all the possible values of the random variable x cannot be enumerated one by one, the random variable at any point in a certain interval on the number line.

2.Discrete random variables.

Let x be a random variable, and if all its possible values are finite or infinite, then x is said to be a discrete random variable.

Classical probability, also known as ex-ante probability, refers to the fact that when the various possible outcomes in a random event and the number of occurrences can be known by deduction or extrapolation, the probability of various possible outcomes can be calculated without any statistical experiments.

Conditional probability refers to the probability of event A occurring under the condition that another event B has already occurred. The conditional probability is expressed as: p(a|b), which is read as "the probability of a under the condition of b".

In probability theory and statistics, expected value (or mathematical expectation, or mean, also known as expectation, or expectation) is the probability of each possible outcome in an experiment of discrete random variables multiplied by the sum of its outcomes.

In probability theory and statistics, a binomial distribution is n independent discrete probability distributions of the number of successes in a non-trial, where the probability of success in each trial is p. Such a single success or failure test is also called the Bernoulli test. In fact, when n = 1, the binomial distribution is the Bernoulli distribution.

The binomial distribution is the basis for the binomial experiment of significant differences.

Bernoulli distribution is also known as "zero-one distribution" and "two-point distribution". The random variable x is said to have a Bernoulli distribution, and the parameter is p (0 0 is a parameter of the distribution, which is often called the rate parameter. That is, the number of times the event occurs per unit of time.

The interval of the exponential distribution is [0, ) and if a random variable x is exponentially distributed, you can write: x exponential( ).

Probability density function:

Skewed distribution refers to the distribution of frequency distribution in which the peak of the limb is located on one side and the tail extends to the other side. It is divided into positive skewed and negatively skewed. The data with skewed distribution can sometimes be converted into a normal distribution after taking the logarithm, and the median is often used to reflect the concentrated trend of the skewed distribution.

In probability theory, a beta distribution, also known as a b distribution, refers to a set of continuous probability distributions defined in the interval (0,1) with two parameters, 0.

Probability density function:

The Weibull distribution is the theoretical basis for reliability analysis and life testing.

Probability density function:

Chi-square distribution is a type of probability distribution commonly used in probability theory and statistics. The sum of squares of k independent standard normally distributed variables obeys a chi-square distribution with degrees of freedom k. Chi-square distribution is a special gamma distribution that is one of the most widely used probability distributions in statistical inference, such as hypothesis testing and the calculation of confidence intervals.

Mathematical Definition: Probability Density Function:

The question is very simple, and the description is cumbersome. Just give the results and see for yourself.

Probability distribution: p=1 20, p=3 20, p=6 20, p=10 20ex=3x1 20+4x3 20+5x6 20+6x10 20=21 4

dx=(3-ex)^2+(4-ex)^2...= Hey, hey, forget it yourself!

Up to 3 tests:

The probability of the first detection of defective products: c(2,1) c(4,1)=2 4=1 2

The probability of not being able to detect defective products for the first time: =1-1 2=1 2;

The drugs that have been tested for the first time will not be put back, and the remaining 3 will be:

The probability of the first detection of defective products, the second detection of defective products is 1 3, and the probability of not detecting defective products for the second time is 2 3;

1) The list is as follows (+ means **, x means defective product, and number means probability):

1...2...3...Total probability. Frequency.

x(1/2)..x(1/3)..1/6...2

x(1/2)..2/3)..x(1/2)..1/6...3

x(1/2)..2/3)..1/2)..1/6...3

1/2)..x(2/3)..1/2)..1/6...3

1/2)..x(2/3)..x(1/2)..1/6...3

2 times: 1 6 + 1 6 = 1 3;

3 times: 1 6 4 = 2 3;

2) More than 2 times, that is, 3 times, probability 2 3;

3) Distribution Function:

p=0，（x<2）

p=1/3,（2≤x<3）

p=1,(x≥3）

The probability of taking out ** for the first time is 3 (3+3)=1 2 --end. No defects Probability 1 2

The probability of taking out a defective product for the first time is 3 (3+3)=1 2

The probability of taking out ** for the second time is 4 (4+2)=2 3 -- end 1 defective product probability 1 2*2 3= 1 3

The probability of taking out the defective product for the second time is 2 (4+2)=1 3

The probability of taking out ** for the third time is 5 (5+1)=5 6 --end 2 defective products probability 1 2*1 3*5 6= 5 36

The probability of taking out the defective product for the third time is 1 (5+1)=1 6

The probability of taking out ** for the fourth time is 6 6 = 1 -- end 3 defective products Probability 1 2*1 3*1 6*1 = 1 36

The probability of taking out the defective product for the first time is 3 (3+3)=1 2, the probability of taking out the defective product for the second time is 2 (4+2)=1 3, the probability of taking out the defective product for the third time is 1 (5+1)=1 6, and the probability of taking out the defective product for the fourth time is 0 6=0

The distribution rate of the defective products is as follows

1st 2nd 3rd 4th 1 2 1 3 1 6 0