1 Tianzaixin Mantou Shop

There is a steamed bun shop downstairs in the company. It is open every morning from 6:00 to 10:00.

The boss counted the steamed buns sold every day for a week (for the convenience of calculation and explanation, the data was reduced):

The mean is:

It is reasonable to say that the mean is a good choice (see How to understand the least squares method ?), but if you prepare 5 steamed buns every day, from the statistical table, at least two days are not enough to sell, and 40% of the time is not enough to sell:

You "Sweet Heart Mantou Shop" is not Xiaomi, what kind of hunger marketing are you doing? Of course, the boss also knew this, so he picked up a pen and paper to start thinking.

2 Thoughts from the boss

The boss tries to abstract the business hours into a line segment and use this time $T$ to represent:

Then put the three steamed buns on Monday ("Sweet in the Heart steamed buns", steamed buns with folds) on the line segment according to the sales time:

Divide it $T$ into four time periods:

At this time, in each time period, either (one) steamed bun was sold or not:

At each time period, it is a bit like flipping a coin, either heads (sold) or tails (no sell).

$T$ The probability of selling 3 steamed buns within a period of time is the same as the probability of tossing a coin 4 times (4 time periods), of which 3 heads (selling 3).

Such probabilities are computed from the binomial distribution as:

However, if you put Tuesday's seven steamed buns on the line segment, it is not enough to divide into four segments:

It can be seen from the figure that in each time period, there are 3 sold, 2 sold, and 1 sold, so it is no longer simply "sold, not sold". The binomial distribution cannot be applied.

Solving this problem is also very simple, divide it $T$ into 20 time periods, then each time period becomes a coin flip:

In this way, $T$ the probability of selling 7 steamed buns is (equivalent to 20 coin tosses and 7 positives):

In order to ensure that only "sell, not sell" will happen within a period of time, simply divide the time $n$ into parts :

The thinner the better, expressed by the limit:

To be more abstract, the probability of $T$ selling $k$ a steamed bun within a certain period of time is:

3 calculation of p

"Then," the boss tapped his pen on the table, "there is only one question left, $p$ how to find the probability?"

Under the above assumptions, the problem has been transformed into a binomial distribution. The expectation of the binomial distribution is:

So:

4 Poisson distribution

Once you have $p=\frac{\mu }{n}$ it, you have:

Let's calculate this limit:

in:

so:

The above is the probability density function of the Poisson distribution $T$ , that is to say, the probability of selling $k$ a steamed bun within a certain time is:

In general, we'll change a notation, let $\mu = \lambda$ , so:

This is the probability density function of the Poisson distribution in the textbook.

5 Solutions to the problems of steamed bun shops

The boss is still frowning, don't you know $\mu$ ?

It doesn't matter, the sample mean was not calculated just now:

It can be approximated with:

then:

The curve that draws the probability density function is:

It can be seen that if 8 steamed buns are prepared every day, the probability of selling enough is to add up the probabilities of the first 8:

In this way, 93% of the cases are sufficient, and occasionally selling out of stock can also help the brand image.

The boss counted the sweat on his forehead, "Then it's settled!"

6 Binomial and Poisson distributions

In view of the relationship between the binomial distribution and the Poisson distribution, it is natural to get an inference that when the binomial distribution $p$ is small, the two are relatively close:

7 summary

This story tells us that we must study hard, or we will not be able to sell steamed buns in the future.

There are many more Poisson distributions in life. For example, the half-life in physics, we only know what is the expected time for half of the material to decay, but because of the uncertainty principle , we have no way to know which atom will decay when? So it can be calculated using Poisson distribution.

There are also issues such as traffic planning and so on.

[Probability and Statistics] Poisson Distribution (1)