4.10.2 Calculating Poisson Probabilities

Throughout this section we will use the random variable . 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 
  • 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
  • 
    • where e is Euler's constant
    • and 

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 
  • Some calculators can only find 
    • 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 
  • The upper value - this is the value b
    • This can be a very large number (9999...) in the case 
  • The 'λ' value - the mean number of occurrences (m)

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

• To calculate P(X ≤ x) just enter x into the cumulative distribution function
• To calculate P(X < x) use:
  • which works when X is a Poisson random variable
    • P(X < 5) = P(X ≤ 4)

• To calculate P(X > x) use:
  • which works for any random variable X 
    • P(X > 5) = 1 - P(X ≤ 5)
• To calculate P(X ≥ x) use:
  •  which works when X is a Poisson random variable
    • P(X ≥ 5) = 1 - P(X ≤ 4)
• To calculate P(a ≤ X ≤ b) use:
  •  which works when X is a Poisson random variable
    • P(5 ≤ X ≤ 9) = P(X ≤ 9) - P(X ≤ 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
  • and 
  • 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
  • 
• 
  • P(5 < X ≤ 9) = P(6 ≤ X ≤ 9)
• 
  • P(5 ≤ X < 9) = P(5 ≤ X ≤ 8)
• 
  • P(5 < X < 9) = P(6 ≤ X ≤ 8)