Poisson distribution

Poisson distribution

Lessons

The Poisson Distribution is an approximation to the Binomial Distribution.

Recall:
P(x)=nCxpx(1p)nxP(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)=eλP(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

    c)
    Compare your previous two answers


  • 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?