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MindMap Gallery In-depth Introduction to Statistics-Geometric distribution, binomial distribution and Poisson distribution

In-depth Introduction to Statistics-Geometric distribution, binomial distribution and Poisson distribution

"In-depth Introduction to Statistics"-Geometric distribution, binomial distribution and Poisson distribution. Spend the least time to understand the statistical knowledge in data analysis. This is part 7, finding special probability distribution~

Edited at 2022-07-29 16:31:53

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In-depth Introduction to Statistics-Geometric distribution, binomial distribution and Poisson distribution

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Outline

7. Special probability distribution (discrete)

geometric distribution

Application conditions

conduct a series of independent trials;

Each trial either succeeds or fails, and each trial has the same probability of success;

Q: How many trials are needed to achieve the first "success"?

Number of trials required to achieve first success

Probability of success for a single trial

probability

independent

If you are skiing, the probability of successfully reaching the bottom of the slope without any accidents in any trial is 0.2

Probability equation

The probability of achieving the first success on the rth trial

Failure probability: q=1-p

In order to succeed in the r-th experiment, it must first fail (r-1) times

It takes more than r trials to obtain the probability of success for the first time

That is, the first r trials must fail.

There is no need for p, because we do not need to know exactly which experiment is successful, we only need that the number of experiments must be greater than r

It needs to be tested r times or less than r times, the probability of first success

Expectation / Variance

E(X)=1/p

The probability of success in a single trial is p, which can be understood as one of the 1/p attempts tends to succeed.

Expect success in 1/p attempts

geometric distribution shape

When r=1, P(X=r) reaches the maximum value, and as r increases, P(X=r) gradually decreases.

mode

The mode of any geometric distribution is always 1 because 1 is the number with the greatest probability

Most likely scenario: Success after only one try

Case

20% of cereal boxes contain free toys, one in each box. The probability of opening less than 4 cereal boxes to get the first free toy is 0.488

X～Geo(0.2) P(X<=3)

binomial distribution

Application conditions

conduct a series of independent trials;

Each trial either succeeds or fails, and each trial has the same probability of success;

Limited number of trials (difference from geometry)

Find: How many times can you succeed in n trials (the probability of n independent repeated trials)

number of successes in n trials

Probability of success for a single trial

Probability equation

Combination: Number of selection ways to select r objects from n objects (no need to know the exact selection order)

Expectation / Variance

E(X)=np

Var(X) =npq

Single test E(x)=p,Var(x)=pq

geometric distribution shape

According to different values of n and p, the shape of the binomial distribution will change.

The closer p is to 0.5, the more symmetrical the graph is.

When p is less than 0.5, the graph is skewed to the right; when p is greater than 0.5, the graph is skewed to the left.

mode

p=0.5 and n is an even number, the mode is np

p=0.5 and n is an odd number, the mode is located at the two values on the left and right sides of np

Other n\p require repeated trial calculations

Case

Someone is bowling and the probability of him knocking down all the pins is 0.3. If he can throw the ball 10 times, what is the probability of knocking down all the pins in 3 times? 0.382

P(X<3)= P(X=0) P(X=1) P(X=2)

P(X=0)=

There are 5 questions in the game. The probability of answering each question correctly is 0.25. What is the probability of answering 2 or 3 questions correctly? 0.3519

P（X=2） P（X=3）

Poisson distribution

Application conditions

A single event occurs randomly and independently within a given interval (the given interval can be time or space, for example, it can be a week or a mile)

It is known that the average number of occurrences of events in a given interval, or the occurrence rate, is a finite value

Find: the number of events occurring within a given interval

The number of occurrences of events in a given interval

incidence

random

The average number of popcorn machine failures per week is 3.4 times, but the actual number of failures is not fixed. It may break frequently in the next week, or it may work fine all the time.

probability formula

Expectation / Variance

E(X)=

Var(X) =

geometric distribution shape

If lambda is small, the distribution will be skewed to the right. As lambda becomes larger, the distribution will become more symmetrical.

lambda is an integer and has two modes,

lambda is a non-integer, mode

Case

A bus will stop once every 15 minutes on average. The probability of no bus appearing within 15 minutes is 0.368

X～Po(1) P(X=0)

special

Poisson distribution for two independent events

P(X Y) = P(X) P(Y) E(X Y) = E(X) E(Y)

If both X and Y conform to the Poisson distribution, then X Y also conforms to the Poisson distribution. The distribution of X and Y can be used to find the probability of X Y

Approximate binomial distribution

Condition: n of X～B(n,p) is large enough and p is small enough, it can be approximately regarded as X～Po(np)

n is large, it is difficult to calculate the combination

When n>50 and p less than 0.1 (typical approximation)