Data Science & Statistics Tutorial: The Poisson Distribution

00:05:08
https://www.youtube.com/watch?v=BbLfV0wOeyc

Resumo

TLDRThe lecture discusses the Poisson Distribution, which is defined by a single parameter, lambda, representing the average frequency of events in a specific interval. It explains how to calculate the probability of a certain number of occurrences using the Poisson probability function, which involves Euler's number and factorials. The expected value and variance of the distribution are both equal to lambda, highlighting the distribution's elegant statistical properties. An example is provided to illustrate the application of the Poisson Distribution in real scenarios.

Conclusões

  • 📊 Poisson Distribution is defined by a single parameter, lambda.
  • 🔍 It measures the frequency of events in a specific interval.
  • 📈 The probability function involves Euler's number and factorials.
  • 💡 Expected value and variance are both equal to lambda.
  • 🧮 Use the formula: p(y) = (lambda^y * e^(-lambda)) / y!.
  • 📅 Example: Calculate the likelihood of 7 questions when average is 4.
  • 🔗 Joint probability is used for intervals in Poisson Distribution.

Linha do tempo

  • 00:00:00 - 00:05:08

    In this lecture, we explore the Poisson Distribution, characterized by a single parameter, lambda. It focuses on the frequency of events occurring within a specific interval rather than the probability of a single event. For instance, if a firefly lights up 3 times in 10 seconds on average, we can use the Poisson Distribution to determine the likelihood of it lighting up 8 times in 20 seconds. The distribution graph starts at 0 and has no upper limit on occurrences. An example illustrates this: if students typically ask 4 questions per day but asked 7 yesterday, we can calculate the probability of receiving exactly 7 questions using the Poisson probability function. The formula involves lambda raised to the power of y, multiplied by Euler's number raised to the power of negative lambda, divided by y factorial. After calculating, we find a 6% chance of receiving exactly 7 questions. Additionally, the expected value and variance of the Poisson Distribution are both equal to lambda, showcasing the distribution's elegant statistical properties. Finally, to compute the probability of an interval, we find the joint probability of all individual elements within that interval.

Mapa mental

Vídeo de perguntas e respostas

  • What is the Poisson Distribution?

    The Poisson Distribution models the frequency of events occurring in a specific interval.

  • What does lambda represent in the Poisson Distribution?

    Lambda represents the average number of occurrences in a given time period.

  • How do you calculate the probability in a Poisson Distribution?

    Use the formula: p(y) = (lambda^y * e^(-lambda)) / y!.

  • What is Euler's number?

    Euler's number, approximately 2.72, is a constant used in the Poisson formula.

  • What is the expected value in a Poisson Distribution?

    The expected value is equal to lambda.

  • What is the variance in a Poisson Distribution?

    The variance is also equal to lambda.

  • How do you find the probability of an interval in a Poisson Distribution?

    Calculate the joint probability of all individual elements within the interval.

  • What is an example of using the Poisson Distribution?

    Calculating the likelihood of receiving 7 questions in a day when the average is 4.

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Legendas
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Rolagem automática:
  • 00:00:03
    Hello again!
  • 00:00:05
    In this lecture we are going to discuss the Poisson Distribution and its main characteristics.
  • 00:00:10
    For starters, we denote a Poisson distribution with the letters “Po” and a single value
  • 00:00:16
    parameter - lambda.
  • 00:00:19
    We read the statement below as “Variable “Y” follows a Poisson distribution with
  • 00:00:24
    lambda equal to 4”.
  • 00:00:27
    Okay!
  • 00:00:29
    The Poisson Distribution deals with the frequency with which an event occurs in a specific interval.
  • 00:00:35
    Instead of the probability of an event, the Poisson Distribution requires knowing how
  • 00:00:40
    often it occurs for a specific period of time or distance.
  • 00:00:45
    For example, a firefly might light up 3 times in 10 seconds on average.
  • 00:00:51
    We should use a Poisson Distribution if we want to determine the likelihood of it lighting
  • 00:00:56
    up 8 times in 20 seconds.
  • 00:01:00
    The graph of the Poisson distribution plots the number of instances the event occurs in
  • 00:01:04
    a standard interval of time and the probability for each one.
  • 00:01:08
    Thus, our graph would always start from 0, since no event can happen a negative amount
  • 00:01:14
    of times.
  • 00:01:15
    However, there is no cap to the amount of times it could occur over the time interval.
  • 00:01:20
    Okay, let us explore an example.
  • 00:01:24
    Imagine you created an online course on probability.
  • 00:01:28
    Usually, your students ask you around 4 questions per day, but yesterday they asked 7.
  • 00:01:36
    Surprised by this sudden spike in interest from your students, you wonder how likely
  • 00:01:40
    it was that they asked exactly 7 questions.
  • 00:01:44
    In this example, the average questions you anticipate is 4, so lambda equals 4.
  • 00:01:51
    The time interval is one entire work day and the singular instance you are interested in
  • 00:01:55
    is 7.
  • 00:01:57
    Therefore, “y” is 7.
  • 00:02:00
    To answer this question, we need to explore the probability function for this type of
  • 00:02:05
    distribution.
  • 00:02:07
    Alright!
  • 00:02:09
    As you already saw, the Poisson Distribution is wildly different from any other we have
  • 00:02:14
    gone over so far.
  • 00:02:16
    It comes without much surprise that its probability function is much different from anything we
  • 00:02:20
    have examined so far.
  • 00:02:22
    The formula looks as follows: “p of y, equals, lambda to the power of
  • 00:02:28
    y, times the Euler’s number to the power of negative lambda, over y factorial.
  • 00:02:36
    Before we plug in the values from our course-creation example, we need to make sure you understand
  • 00:02:41
    the entire formula.
  • 00:02:42
    Let’s refresh your knowledge of the various parts of this formula.
  • 00:02:47
    First, the “e” you see on your screens is known as Euler’s number or Napier’s
  • 00:02:52
    constant.
  • 00:02:53
    As the second name suggests, it is a fixed value approximately equal to 2.72.
  • 00:03:00
    We commonly observe it in physics, mathematics and nature, but for the purposes of this example
  • 00:03:05
    you only need to know its value.
  • 00:03:07
    Secondly, a number to the power of “negative n”, is the same as dividing 1 by that number
  • 00:03:14
    to the power of n.
  • 00:03:16
    In this case, “e to the power or negative lambda” is just “1 over, e to the power
  • 00:03:22
    of lambda”.
  • 00:03:24
    Right!
  • 00:03:26
    Going back to our example, the probability of receiving 7 questions is equal to “4,
  • 00:03:31
    raised to the 7th degree, multiplied by “E” raised to the negative 4, over 7 factorial,”.
  • 00:03:40
    That approximately equals 16384, times 0.183, over 5040, or 0.06.
  • 00:03:51
    Therefore, there was only a 6% chance of receiving exactly 7 questions.
  • 00:03:59
    So far so good!
  • 00:04:01
    Knowing the probability function, we can calculate the expected value.
  • 00:04:05
    By definition, the expected value of Y, equals the sum of all the products of a distinct
  • 00:04:10
    value in the sample space and its probability.
  • 00:04:13
    By plugging in, we get this complicated expression.
  • 00:04:17
    Eventually, we get that the expected value is simply lambda.
  • 00:04:22
    Similarly, by applying the formulas we already know, the variance also ends up being equal
  • 00:04:27
    to lambda.
  • 00:04:29
    Both the mean and variance being equal to lambda serves as yet another example of the
  • 00:04:34
    elegant statistics these distributions possess and why we can take advantage of them.
  • 00:04:41
    Great job, everyone!
  • 00:04:42
    Now, if we wish to compute the probability of an interval of a Poisson distribution,
  • 00:04:47
    we take the same steps we usually do for discrete distributions.
  • 00:04:52
    We find the joint probability of all individual elements within it.
Etiquetas
  • Poisson Distribution
  • lambda
  • Euler's number
  • probability function
  • expected value
  • variance
  • statistics
  • discrete distributions
  • frequency of events
  • interval probability