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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.

  • Usually, your students ask you around 4 questions per day, but yesterday they asked 7.

  • 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!


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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