Continuous density function properties -> Solve three problems (find the unknown coefficients, find the probability, find the density function)
Distribution function -> solve three problems
Common distributions (the ones in the last lesson)
three:
Seven discrete (continuous) type questions: (distribution law (coefficient of determination)), probability, marginal distribution (density), independence, conditional distribution (density), function distribution, covariance (correlation coefficient)
Four:
Mathematical expectation, variance (calculation, common distribution, analysis)
Chebyshev's inequality
Two-dimensional - correlation and independence, covariance of two variables
five:
Central Limit Theorem
first lesson
1.1 No-replacement questions (classical probability model)
1.2 Questions with Replacement
1.3 Questions that require drawing
1.4 Total Probability Formula
The probability of two independent events A and B occurring at the same time is P(AB) = P(A) * P(B)
1.5 Bayesian formula
1.6 Event Probability (Relational Operation/Conditional Probability)
addition:
Subtraction:
Multiplication and division:
Independent events do not affect each other.
Second lesson
2.1 Given one of the distribution function Fx(x) and the density function fx(x), find the other
2.2 Given one of Fx(x) and fx(x), find P
The equal sign in P has no effect here. The presence or absence of the x subscript in F or f has no effect on itself.
2.3 Fx(x) or fx(x) contains unknowns. Find the unknowns.
Several formulas for standardization.
2.4 Finding the distribution law
The distribution sequence is the distribution law.
Problems like rolling dice are sequencing problems (A).
2.5 Given a distribution sequence containing unknown numbers, find the unknown numbers
Given the following distribution, find the value of k.
Lesson Three
3.1 Given the distribution of X, find the distribution of Y
The following can also be written:
Fourth lesson
4.1 Find the probability of uniform distribution
4.2 Find the probability of Poisson distribution
4.3 Find the probability of binomial distribution
4.4 Find the probability of exponential distribution
4.5 Normal distribution, find the probability
Standard normal distribution, N(0, 1).
fifth lesson
5.1 Given a two-dimensional discrete distribution law, what is it?
5.2 Given a two-dimensional discrete distribution law, determine independence
5.3 Given F(x, y), find f(x, y)
5.4 Distribution Function F(x) and Probability Density f(x) of a Continuous Two-Dimensional Variable
5.4.1 Finding the probability density f(x) and probability
5.4.2 Finding the Undetermined Coefficients and Distribution Function F(x)
What if there are three unknown terms? Use segmentation to find out the continuity.
Lesson Six
6.1 Finding the Marginal Distribution Function
6.2 Finding the Marginal Density Function
6.3 Determining the independence of continuous two-dimensional variables
fx(x) and fy(y) have already been found in the previous problem type.
6.4 Two-dimensional discrete random distributions (joint, marginal, conditional distributions, and independence)
6.5 Two-dimensional continuous random distributions (joint, marginal, conditional density, and independence)
By norm, the probability of the rectangular area is 1.
The discrete type is to find the distribution law.
Lesson Seven
7.1 Finding the Discrete Expectation E(x)
7.2 Finding the expectation E(x) of a continuous form
7.3 Given Y = g(x), find E(y)
7.4 Finding the Variance D(x)
7.5 Performing complex operations based on the properties of E(x) and D(x)
7.6 Comprehensive Problems on E(X), D(X) and Various Distributions
Lesson Eight
8.1 Covariance Cov, density coefficient Pxy, variance D and related questions
Discrete:
Continuous type:
If P(rou) is not equal to 0, then X and Y are correlated.
8.2 Using Chebyshev's inequality to find probability
8.3 Multiple independent and identically distributed numbers, what is the probability of the sum?
Lesson 9
9.1 Finding the Expectation of a Discrete Form
9.2 Finding the Expectation of a Continuous Form
9.3 Given Y=g(x), find E(Y)
9.4 Finding the Variance D(x)
9.5 Complex operations based on the properties of E(x) and D(x)
9.6 Comprehensive Problems on E(x), D(x) and Various Distributions
0-1 distribution: E(x) = p; D(x) = p(1 - p)
The binomial distribution is also a Bernoulli distribution (independent, n repeated trials, each time there are only two results, A and not A).
Lesson 10
Central Limit Theorem
n variables, independent, identically distributed
After normalization, we get the standard normal distribution:
