Calculating Poisson Probabilities (DP IB Applications & Interpretation (AI): HL): Revision Note

Calculating Poisson probabilities

Throughout this section we will use the random variable $X~\mathrm{Po}\left(m\right)$. 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. However the value of m can be any real positive value (it doesn't need to be an integer).

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 $\mathrm{P}(X=x)$

  • 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

  • $\mathrm{P}(X=x)=\frac{{\mathrm{e}}^{-m}{m}^x}{x!}$

    • where e is Euler's constant

    • $x!=x\times \left(x-1\right)\times \dots \times 2\times 1$ and $0!=1$

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 $\mathrm{P}(a\le X\le b)$

  • Some calculators can only find $\mathrm{P}(X\le b)$

    • 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 $\mathrm{P}(X\le b)$

  • The upper value - this is the value b

    • This can be a very large number (9999... or 10⁹⁹) in the case $\mathrm{P}(X\ge a)$

  • 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:

  • $\mathrm{P}(X<x)=\mathrm{P}(X\le x-1)$ which works when X is a Poisson random variable

    • P(X < 5) = P(X ≤ 4)

• To calculate P(X > x) use:

  • $\mathrm{P}(X>x)=1-\mathrm{P}(X\le x)$ which works for any random variable X 

    • P(X > 5) = 1 - P(X ≤ 5)

• To calculate P(X ≥ x) use:

  • $\mathrm{P}(X\ge x)=1-\mathrm{P}(X\le x-1)$ which works when X is a Poisson random variable

    • P(X ≥ 5) = 1 - P(X ≤ 4)

• To calculate P(a ≤ X ≤ b) use:

  • $\mathrm{P}(a\le X\le b)=\mathrm{P}(X\le b)-\mathrm{P}(X\le a-1)$ 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

  • $\mathrm{P}(X<x)=\mathrm{P}(X\le x-1)$ and $\mathrm{P}(X>x)=\mathrm{P}(X\ge x+1)$

  • 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

  • $\mathrm{P}(X\le b)=\mathrm{P}(0\le X\le b)$

• $\mathrm{P}(a<X\le b)=\mathrm{P}(a+1\le X\le b)$

  • P(5 < X ≤ 9) = P(6 ≤ X ≤ 9)

• $\mathrm{P}(a\le X<b)=\mathrm{P}(a\le X\le b-1)$

  • P(5 ≤ X < 9) = P(5 ≤ X ≤ 8)

• $\mathrm{P}(a<X<b)=\mathrm{P}(a+1\le X\le b-1)$

  • P(5 < X < 9) = P(6 ≤ X ≤ 8)