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)
- The lower value - this is the value a
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)
- which works when X is a Poisson random variable
- To calculate P(X > x) use:
- which works for any random variable X
- P(X > 5) = 1 - P(X ≤ 5)
- which works for any random variable X
- To calculate P(X ≥ x) use:
- which works when X is a Poisson random variable
- P(X ≥ 5) = 1 - P(X ≤ 4)
- which works when X is a Poisson random variable
- 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)
- which works when X is a Poisson random variable
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)
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- P(5 ≤ X < 9) = P(5 ≤ X ≤ 8)
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- P(5 < X < 9) = P(6 ≤ X ≤ 8)
Worked example
The random variables and are independent. Find: