Ruan Teacher Pausong Distribution

Poisson distribution and American shootings

Nanyi

Date: January 8, 2013

Last December, 28 people were killed in a school shooting in Connecticut State, USA.

Data show that from 1982 to 2012, the United States has a total of 62 (large-scale) shooting cases. Of these, 7 occurred in 2012, the highest number of years.

Is it a coincidence that there were so many shootings last year, or is it a sign that the U.S. is deteriorating?

A few days ago, I saw an interesting article using "Poisson's distribution" (Poisson distribution) to determine whether 7 shootings in the same year were coincidental.

Let's start with an example of what "Poisson distribution" is.

A small grocery store is known to sell 2 cans of fruit per week on average. What is the best stock of the fruit can in this shop?

Assuming that there are no seasonal factors, it can be approximated that this problem satisfies the following three conditions:

(1) The customer buys the fruit can is the small probability event.

(2) Customers who buy canned fruit are independent and do not affect each other.

(3) The probability of a customer buying a canned fruit is stable.

Statistically, as long as a certain event satisfies the above three conditions, it obeys the "Poisson distribution".

The formula for Poisson distribution is as follows:

The meaning of each parameter:

P: The probability of selling K cans per week.

X: The sales variable of canned fruit.

The value of the k:x (0,1,2,3 ...).

λ: The average sales of canned fruits per week is a constant, this is entitled 2.

According to the formula, the distribution of weekly sales is calculated:

As can be seen from the table above, if the stock is 4 cans, 95% of the probability will not be out of stock (on average every 19 weeks); If the inventory of 5 cans, 98% of the probability will not be out of stock (on average 59 weeks occurs).

Now, let's go back and look at the American shootings.

Assume that they meet the three conditions of a "Poisson distribution":

(1) The shooting case is a small probability event.

(2) The shootings are independent and do not affect each other.

(3) The probability of occurrence of shooting is stable.

Clearly, the third condition is the key. If it is established, it shows that the law and order in the United States has not deteriorated, and if not, it indicates that the probability of the shooting is unstable, is improving, and the United States security deterioration.

According to the information, the 1982--2012 shooting case is distributed as follows:

The average of 2 shootings per year is calculated, so λ= 2.

, the blue bar is the actual observed value, and the red dashed line is the expected value of the theory. As you can see, the observations are quite close to the expected value.

We use the "chi-square test" (chi-square test) to verify that there is a significant difference between the observed and expected values.

Chi-Square Statistic =σ[(observation-expected value) ^ 2/expected value]

Calculated, the chi-square statistic equals 9.82. After tabular, the chi-square distribution threshold of confidence level 0.90 and Freedom 7 is 12.017. Therefore, the chi-square statistic is less than the critical value, which indicates that there is no significant difference between the observed and expected values of the shootings. Therefore, it is acceptable to assume that the probability of a shooting is stable, that is to say, statistics do not get the conclusion that the law of the United States is deteriorating.

However, it must also be seen that Chi-square statistic 9.82 is close to the critical value, and P-value is only 0.18. In other words, we are only 82% confident that there is no deterioration in U.S. security, and 18% of us may be wrong, and the law of the United States is actually deteriorating. This, therefore, will need to see whether there are still a large number of shootings occurring in the next two years. If it does happen, the Poisson distribution will not be established.

Ruan Teacher Pausong Distribution