Poisson distribution

Poisson distribution

Lessons

The Poisson Distribution is an approximation to the Binomial Distribution.

Recall:
• P(x)=nCxpx(1−p)n−xP(x)= {_n}C_{x}\;p^x(1-p)^{n-x}
nn: number of trials
xx: number of success in n trials
pp: probability of success in each trial
P(x)P(x): probability of getting x successes (out of n trials)
• μ=np\mu=np
Now:
• μ=λ=np\mu=\lambda=np

Poisson Distribution:
P(x)=e−λP(x)=e^{-\lambda} λxx!\frac{\lambda^x}{x!}
• poissonpdf (λ,x)(\lambda,x)
• poissoncdf (λ,x)(\lambda,x)
  • 1.
    Determining the Poisson Distribution
    The number of meteors that hit the earth in a given day is modelled by a Poisson Distribution with λ=4\lambda=4. What is the probability that 5 meteors hit the earth in a day?
  • 2.
    When making a video I typically make 1 error for every 20 minutes of video time. If I make 45 minutes of video what is the probability that I make 3 errors?
  • 3.
    In a particular community the average person survives to age 100 with probability 0.005 (which is equivalent to 0.5%). If this community has 2,000 people, then what is the probability that 15 people in this community survive to age 100 using;
    a)
    The Binomial Distribution

    b)
    The Poisson Distribution

  • 4.
    Cumulative Poisson Distribution
    On U.S. route 66 (an American highway) every car that travels this whole route has a probability of p=0.0002p=0.0002 of getting into a car accident. A total of 10,000 cars drive this route every month. What is the probability that there are fewer than 3 car accidents in a month?
  • 5.
    Determining the Poisson Distribution using Calculator Commands
    A fair coin is flipped 10 times, what is the probability using the Poisson Distribution commands on your calculator find,
    a)
    The probability that heads comes up 5 times?

    b)
    The probability that heads comes up 5 or less times?

    c)
    The probability that heads come up more than 7 times?

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4.
Discrete Probabilities
4.1
Probability distribution - histogram, mean, variance & standard deviation
4.2
Binomial distribution
4.3
Mean and standard deviation of binomial distribution
4.4
Poisson distribution
4.5
Geometric distribution
4.6
Negative binomial distribution
4.7
Hypergeometric distribution
4.8
Properties of expectation
Statistics
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