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• Hello again!

• In this lecture we are going to discuss the Poisson Distribution and its main characteristics.

• For starters, we denote a Poisson distribution with the lettersPoand a single value

• parameter - lambda.

• We read the statement below asVariable “Y” follows a Poisson distribution with

• lambda equal to 4”.

• Okay!

• The Poisson Distribution deals with the frequency with which an event occurs in a specific interval.

• Instead of the probability of an event, the Poisson Distribution requires knowing how

• often it occurs for a specific period of time or distance.

• For example, a firefly might light up 3 times in 10 seconds on average.

• We should use a Poisson Distribution if we want to determine the likelihood of it lighting

• up 8 times in 20 seconds.

• The graph of the Poisson distribution plots the number of instances the event occurs in

• a standard interval of time and the probability for each one.

• Thus, our graph would always start from 0, since no event can happen a negative amount

• of times.

• However, there is no cap to the amount of times it could occur over the time interval.

• Okay, let us explore an example.

• Imagine you created an online course on probability.

• Surprised by this sudden spike in interest from your students, you wonder how likely

• it was that they asked exactly 7 questions.

• In this example, the average questions you anticipate is 4, so lambda equals 4.

• The time interval is one entire work day and the singular instance you are interested in

• is 7.

• Therefore, “y” is 7.

• To answer this question, we need to explore the probability function for this type of

• distribution.

• Alright!

• As you already saw, the Poisson Distribution is wildly different from any other we have

• gone over so far.

• It comes without much surprise that its probability function is much different from anything we

• have examined so far.

• The formula looks as follows: “p of y, equals, lambda to the power of

• y, times the Euler’s number to the power of negative lambda, over y factorial.

• Before we plug in the values from our course-creation example, we need to make sure you understand

• the entire formula.

• Let’s refresh your knowledge of the various parts of this formula.

• First, the “e” you see on your screens is known as Euler’s number or Napier’s

• constant.

• As the second name suggests, it is a fixed value approximately equal to 2.72.

• We commonly observe it in physics, mathematics and nature, but for the purposes of this example

• you only need to know its value.

• Secondly, a number to the power ofnegative n”, is the same as dividing 1 by that number

• to the power of n.

• In this case, “e to the power or negative lambdais just “1 over, e to the power

• of lambda”.

• Right!

• Going back to our example, the probability of receiving 7 questions is equal to “4,

• raised to the 7th degree, multiplied by “E” raised to the negative 4, over 7 factorial,”.

• That approximately equals 16384, times 0.183, over 5040, or 0.06.

• Therefore, there was only a 6% chance of receiving exactly 7 questions.

• So far so good!

• Knowing the probability function, we can calculate the expected value.

• By definition, the expected value of Y, equals the sum of all the products of a distinct

• value in the sample space and its probability.

• By plugging in, we get this complicated expression.

• Eventually, we get that the expected value is simply lambda.

• Similarly, by applying the formulas we already know, the variance also ends up being equal

• to lambda.

• Both the mean and variance being equal to lambda serves as yet another example of the

• elegant statistics these distributions possess and why we can take advantage of them.

• Great job, everyone!

• Now, if we wish to compute the probability of an interval of a Poisson distribution,

• we take the same steps we usually do for discrete distributions.

• We find the joint probability of all individual elements within it.

Hello again!

B1 中級

# データサイエンスと統計学のチュートリアル.ポアソン分布 (Data Science & Statistics Tutorial: The Poisson Distribution)

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林宜悉 に公開 2021 年 01 月 14 日