Calculating Poisson Probabilities (DP IB Maths: AI HL)

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Dan

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Dan

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Calculating Poisson Probabilities

Throughout this section we will use the random variable X blank tilde Po invisible function application open parentheses m close parentheses. For a Poisson distribution X, the probability of X taking a non-integer or negative value is always zero. Therefore, any values mentioned in this section for X will be assumed to be non-negative integers. The value of m can be any real positive value.

How do I calculate P(X = x): the probability of a single value for a Poisson distribution?

  • You should have a GDC that can calculate Poisson probabilities
  • You want to use the "Poisson Probability Distribution" function
    • This is sometimes shortened to PPD, Poisson PD or Poisson Pdf
  • You will need to enter:
    • The 'x' value - the value of x for which you want to find straight P left parenthesis X equals x right parenthesis
    • The 'λ' value - the mean number of occurrences (m)
  • Some calculators will give you the option of listing the probabilities for multiple values of x at once
  • There is a formula that you can use but you are expected to be able to use the distribution function on your GDC
    • straight P left parenthesis X equals x right parenthesis equals fraction numerator straight e to the power of negative m end exponent m to the power of x over denominator x factorial end fraction
      • where e is Euler's constant
      • x factorial equals x cross times open parentheses x minus 1 close parentheses cross times horizontal ellipsis cross times 2 cross times 1 and 0 factorial equals 1

How do I calculate P(a X b): the cumulative probabilities for a Poisson distribution? 

  • You should have a GDC that can calculate cumulative Poisson probabilities
    • Most calculators will find straight P left parenthesis a less or equal than X less or equal than b right parenthesis
    • Some calculators can only find straight P left parenthesis X less or equal than b right parenthesis
      • The identities below will help in this case
  • You should use the "Poisson Cumulative Distribution" function
    • This is sometimes shortened to PCD, Poisson CD or Poisson Cdf
  • You will need to enter:
    • The lower value - this is the value a
      • This can be zero in the case straight P left parenthesis X less or equal than b right parenthesis
    • The upper value - this is the value b
      • This can be a very large number (9999...) in the case straight P left parenthesis X greater or equal than a right parenthesis
    • The 'λ' value - the mean number of occurrences (m)

How do I find probabilities if my GDC only calculates P(≤ x)?

  • To calculate P(Xx) just enter x into the cumulative distribution function
  • To calculate P(X < x) use:
    • straight P left parenthesis X less than x right parenthesis equals straight P left parenthesis X less or equal than x minus 1 right parenthesis which works when is a Poisson random variable
      • P(X < 5) = P(≤ 4)

  • To calculate P(X > x) use:
    • straight P left parenthesis X greater than x right parenthesis equals 1 minus straight P left parenthesis X less or equal than x right parenthesis which works for any random variable
      • P(X > 5) = 1 - P(≤ 5)
  • To calculate P(Xx) use:
    • straight P left parenthesis X greater or equal than x right parenthesis equals 1 minus straight P left parenthesis X less or equal than x minus 1 right parenthesis which works when is a Poisson random variable
      • P(X ≥ 5) = 1 - P(≤ 4)
  • To calculate P(a Xb) use:
    • straight P left parenthesis a less or equal than X less or equal than b right parenthesis equals straight P left parenthesis X less or equal than b right parenthesis minus straight P left parenthesis X less or equal than a minus 1 right parenthesis which works when is a Poisson random variable
      • P(5 ≤ ≤ 9) = P(≤ 9) - P(≤ 4)

What if an inequality does not have the equals sign (strict inequality)? 

  • For a Poisson distribution (as it is discrete) you could rewrite all strict inequalities (< and >) as weak inequalities (≤ and ≥) by using the identities for a Poisson distribution
    • straight P left parenthesis X less than x right parenthesis equals straight P left parenthesis X less or equal than x minus 1 right parenthesis and straight P left parenthesis X greater than x right parenthesis equals straight P left parenthesis X greater or equal than x plus 1 right parenthesis
    • For example: P(X < 5) = P(X ≤ 4) and P(X > 5) = P(X ≥ 6)
  • It helps to think about the range of integers you want
    • Identify the smallest and biggest integers in the range
  • If your range has no minimum then use 0
    • straight P left parenthesis X less or equal than b right parenthesis equals straight P left parenthesis 0 less or equal than X less or equal than b right parenthesis
  • straight P left parenthesis a less than X less or equal than b right parenthesis equals straight P left parenthesis a plus 1 less or equal than X less or equal than b right parenthesis
    • P(5 < X ≤ 9) = P(6 ≤ X ≤ 9)
  • straight P left parenthesis a less or equal than X less than b right parenthesis equals straight P left parenthesis a less or equal than X less or equal than b minus 1 right parenthesis
    • P(5 ≤ X < 9) = P(5 ≤ X ≤ 8)
  • straight P left parenthesis a less than X less than b right parenthesis equals straight P left parenthesis a plus 1 less or equal than X less or equal than b minus 1 right parenthesis
    • P(5 < X < 9) = P(6 ≤ X ≤ 8)

Worked example

The random variables X tilde Po left parenthesis 6.25 right parenthesis and Y tilde Po left parenthesis 4 right parenthesis are independent. Find:

i)
straight P left parenthesis X equals 5 right parenthesis,

4-10-2-ib-ai-hl-poisson-prob-a-we-solution

ii)
straight P left parenthesis Y less or equal than 5 right parenthesis,

4-10-2-ib-ai-hl-poisson-prob-b-we-solution

iii)
straight P left parenthesis X plus Y greater than 7 right parenthesis.

4-10-2-ib-ai-hl-poisson-prob-c-we-solution

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.