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MindMap Gallery Discrete random variable mind map

Discrete random variable mind map

This is a mind map about discrete random variables, including the concept of random variables, probability distributions of discrete random variables, numerical characteristics of discrete random variables, commonly used probability distributions, etc.

Edited at 2023-11-03 11:12:31

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Discrete random variable mind map

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Outline

Discrete random variable

4.1 Concept of random variables

Random variable: On the sample space, a basic event is mapped to a real number. Such a mapping is called a random variable. A random variable is a mapping of sample space to real numbers.

The definition domain of random variable X is the sample space, and its value range is the set of real numbers.

If the possible values of a random variable are limited or countable, it is called a discrete random variable, otherwise it is called a continuous random variable.

4.2 Probability distribution of discrete random variables

4.2.1 Probability distribution

In probability, capital letters (such as X) are usually used to represent random variables, and lowercase letters (x) represent the specific values that the random variable may take. For each possible value probability distribution of a random variable

The probability distribution/distribution column of a discrete random variable

P(X=xk)=pk, 0≤pk≤1, the sum of all pk is 1. Each basic event in the sample space has one and only one X value corresponding to it. When X takes all possible values, all basic events can be obtained, which is an inevitable event.

4.2.2 Cumulative distribution function

Cumulative distribution function (cdf): Assume a random variable X, for any value x, the probability value P (X≤x) is called the cumulative distribution function of

The graph of the cumulative distribution function can be used to identify discrete random variables and continuous random variables: discrete type - step function; continuous type - smooth curve

4.2.3 Relationship between probability distribution and frequency distribution

Frequency-empirical distribution/statistical distribution; probability-theoretical distribution/population distribution

Testing the applicability of a model (referring to probability distribution) is achieved by comparing the difference between the frequency distribution and probability distribution of a limited observation sample → Goodness of fit test

4.3 Numerical characteristics of discrete random variables

Population characteristic: a number describing the characteristics of a probability distribution, including mathematical expectation (expectation, population mean), variance, standard deviation, etc.

Mathematical expectation: The expected value E(X) is a weighted average, which can be used as a numerical feature to describe the average level of this random variable.

Variance and standard deviation: Variance represents the degree of dispersion of random variable values from the overall expectation. The larger the value, the more dispersed the values.

4.4 Commonly used probability distributions

4.4.1 Bernoulli distribution

Definition: If the probability distribution of a random variable X is P(X=1)=p, P(X=0)=q, (0<p<1, q=1-p), then

Condition: The result of the test can only be one of two opposite results, and cannot occur at the same time; the two results are complete, p q=1

Expectation, variance, standard deviation

Expect μ=p

variance

standard deviation

4.4.2 Binomial distribution

Definition: X~B(n, p). X represents the number of successes in n-fold Bernoulli trials

nature

0≤pk≤1

Applicable conditions

n experiments are performed under the same conditions, and the observations are independent of each other

Only one of two opposing outcomes will occur in each trial

The probability p of "success" in each trial is the same

Cumulative probability: calculate the probability at any point or any interval

binomial distribution graph

n remains unchanged. When p=0.5, the graph is symmetrical. When p>0.5, the graph is left skewed. When p<0.5, the graph is right skewed.

When p remains unchanged and n increases, the above distributions all approach symmetrical distributions.

The shape of the binomial distribution is determined by the parameters n and p. When np≥5 and nq≥5, the binomial distribution has a good approximation to the normal distribution.

Expectation, Variance and Standard Deviation

Expect μ=np

variance

standard deviation

4.4.3 Multinomial distribution

Definition: X~M(n,p)

Expectation, variance and standard deviation: Similar to the binomial distribution, the outcome events A1, A2,...,Ak are divided into two groups, assuming that the group of interest is group Ai (i=1, 2,...,k) , this group only contains one basic event, and the remaining k-1 events are combined and called the Bi group. Therefore, the expectation, variance, and standard deviation of Ai can be directly given by the binomial distribution.

4.4.4 Poisson distribution

The Poisson distribution is associated with rare events and provides a model for the number of rare events that occur in units of time, area, volume, etc.

definition

basic properties

Applicable conditions

A trial is the number X of times an event occurs in a given unit of time, area, or volume

Stability: The probability of an occurrence in a given unit of time, area, or volume is the same for all units

Independence: The number of events that occur in a certain unit of time, area, or volume is independent of the number of events that occur in other units.

Poisson distribution graph

The shape of the graph changes as the λ value changes. The smaller the value of λ, the more skewed the distribution; as the value of λ increases, the distribution gradually becomes more symmetrical.

When λ ≥ 10, the Poisson distribution can be approximated as a normal distribution.

Expectation, variance and standard deviation

expect

variance

standard deviation

The parameter λ in the Poisson distribution is both the expectation and the variance. If the expectation and variance of a discrete random variable are approximately equal, there is reason to believe that the sample data approximately obeys the Poisson distribution.

Poisson distribution and binomial distribution approximation

An important difference between the Poisson distribution and the binomial distribution is the difference in the number of trials n: in the binomial distribution n is limited, in the Poisson distribution n is large enough, p is small, and np is moderate.

When n is large, p is small, and np is moderate, you can save the trouble of using the binomial distribution calculation process and use the Poisson distribution to approximate the binomial distribution. The binomial distribution at this time can be well approximated by the Poisson distribution with parameter λ = np.