Poisson distribution and exponential distribution: 10-minute tutorials

When I was in college, I always thought statistics was difficult and almost hung up.

After work only found that the difficult is not statistics, but our textbooks are not well written. Compared to advanced mathematics, the concept of statistics is much easier to understand.

Let me give you an example of what is Poisson distribution and exponential distribution? I'm afraid most people are not clear about it.

I can get you to understand these two concepts effortlessly in 10 minutes.

First, Poisson distribution

In daily life, a large number of events are fixed frequency.

• An average of 3 babies born per hour in a hospital
• An average company receives 1 calls every 10 minutes
• Average daily sales of 4 packs of XX brand milk powder in a supermarket
• A website has an average of 2 visits per minute

Their characteristic is that we can estimate the total number of these events, but we have no way of knowing the exact time of occurrence. 3 babies are known to be born per hour, how many will be born in the next one hours?

It is possible to have 6 births at a single birth, or they may not be born. That's what we can't know.

Poisson distribution is a description of the occurrence probability of the event in a certain period of time.

Above is the formula for Poisson's distribution. The left side of the equals sign, p for probability, N for a function relationship, T for time, N for quantity, and the probability of birth of 3 infants within 1 hours, expressed as P (n (1) = 3). To the right of the equal sign, λ indicates the frequency of the event.

For the next two hours, the probability that an infant is not born is 0.25%, which is unlikely to happen.

In the next one hours, at least two babies were born with a probability of 80%.

The shape of the Poisson distribution is probably the following.

It can be seen that, near the frequency, the probability of the occurrence of the event is the highest, and then to the two sides of the symmetrical decline, that becomes larger and smaller is unlikely. 3 Babies are born every hour, which is the most likely outcome, and the more or less they are born, the more unlikely they are.

Second, the index distribution

The exponential distribution is the probability of an event's time interval. The following are exponential distributions.

• The time interval at which the baby was born
• Time interval for calls
• Time interval for milk powder sales
• Time interval for site visits

The formula for exponential distribution can be inferred from the Poisson distribution. If the next baby is to be separated by a time t, it is equivalent to not having any babies born within T.

Conversely, the probability that an event occurs within a time t is 1 minus the value above.

For the next 15 minutes, the probability of a baby being born is 52.76%.

For the next 15 minutes to 30 minutes, there is a 24.92% chance that a baby will be born.

The graph of the exponential distribution is probably the following.

It can be seen that the probability of occurrence of events decreases sharply as the interval becomes longer, exponential attenuation. Think, if the average birth of 3 babies per hour, the above has been counted, the next baby 2 hours before the probability of birth is 0.25%, then the interval of 3 hours, 4 hours interval probability, is not closer to 0?

Iii. Summary

In a nutshell: Poisson distribution is the probability distribution of the number of independent events occurring per unit of time, and the exponential distribution is the probability distribution of the time interval of independent events.

Note that the "independent event", the Poisson distribution and exponential distribution, is the premise that events cannot be correlated, otherwise the above formula cannot be applied.

[description] This article is inspired by Nbviewer documentation.

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Nanyi

Original address: http://www.ruanyifeng.com/blog/2015/06/poisson-distribution.html

(turn) Poisson distribution and exponential distribution: 10-minute tutorial