Home / IB DP Maths 2026, 2027 & 2028 / Application and Interpretation HL / IB Mathematics AI AHL Poisson distribution MAI Study Notes

IB Mathematics AI AHL Poisson distribution MAI Study Notes - New Syllabus

IB Mathematics AI AHL Poisson distribution MAI Study Notes

LEARNING OBJECTIVE

  • Poisson distribution

Key Concepts: 

  • Poisson distribution, its mean and variance.
  • Sum of two independent Poisson distributions has a Poisson distribution.

MAI HL and SL Notes – All topics

The Poisson Distribution – $\text{Po}(m)$

The Poisson Distribution – $\text{Po}(m)$

A Poisson distribution models the number of events $X \in \{0, 1, 2, 3, \dots\}$ occurring in a fixed interval (of time, space, etc.), when these events occur:

  • independently of each other, and
  • at a constant average rate.

The probability mass function is:

$P(X = x) = \frac{e^{-m} m^x}{x!}, \quad x = 0, 1, 2, \dots$

Where:

$m$: mean number of occurrences in the interval
$e \approx 2.718$: Euler’s number

$X \sim \text{Po}(m)$

Example 

A call center receives 2 calls per minute on average. Find the probability of receiving exactly 3 calls in one minute.
▶️ Answer/Explanation
Use the Poisson formula:
$ P(X = 3) = \frac{e^{-2} \cdot 2^3}{3!} $
$= \frac{e^{-2} \cdot 8}{6} \approx \boxed{0.180} $

Mean and Variance

If $X \sim \text{Po}(m)$, then:

$\mathbb{E}(X) = m, \quad \text{Var}(X) = m$

Conditions for a Poisson Distribution

1. Independence: Events in disjoint intervals are independent.
2. Uniform Rate: The average rate $m$ is constant over the interval.
3. Single Events: Only one event can occur at a time (no simultaneous events).
4. Rare Events: Poisson is most appropriate when events are relatively rare.

Non-Overlapping Intervals

If we examine events in different intervals, we assume:

The number of events in each interval follows a Poisson distribution.
Events in non-overlapping intervals are statistically independent.

Example 

The probability that 3 calls occur in the first minute and 4 in the second minute:
▶️ Answer/Explanation
Since intervals are independent:
$ P(X=3 \text{ and } Y=4) = P(X=3) \cdot P(Y=4) $
$ = 0.1804 \cdot 0.0902 = \boxed{0.0163} $
Example 

A call center receives 2 calls per minute on average. Use your calculator to find the probability of receiving exactly 3 calls in one minute.
▶️ Answer/Explanation
We want to calculate \( P(X = 3) \) for \( X \sim \text{Po}(2) \).

Using TI-Nspire:
• Press menu → 6: Probability → 5: Distributions → D: Poisson PDF
• Enter: μ = 2, x = 3
• Result: 0.180

Using Casio fx-CG50:
• Go to MENU → STAT or DISTR if available
• Select Poisson P(3, 2)
• Result: 0.180

Final Answer: \( \boxed{0.180} \)

Sum of Independent Poisson Distributions

The sum of two independent Poisson random variables is also a Poisson random variable. If \( X \) and \( Y \) are independent Poisson random variables with parameters \( \lambda_x \) and \( \lambda_y \), respectively, then:

$ X \sim \text{Po}(m), \quad Y \sim \text{Po}(n), \quad X \text{ and } Y \text{ independent} $

$ \Rightarrow X + Y \sim \text{Po}(m + n) $

Example

Call center A receives an average of 2 calls per hour, and B receives 3 calls per hour. Find the probability that they receive a total of 6 calls in an hour.

▶️Answer/Explanation

Let \( X \sim \text{Po}(2) \), \( Y \sim \text{Po}(3) \). Since X and Y are independent:

$X + Y \sim \text{Po}(2 + 3) = \text{Po}(5)$

Then:

$ P(X + Y = 6) = \frac{5^6 e^{-5}}{6!} \approx 0.146 $

Cumulative Distribution Function (CDF)

$ P(X \leq x), \quad P(X < x), \quad P(X \geq x) $

We use the CDF, accessible via GDC. Use the Poisson CDF or Poisson Cumulative function to evaluate cumulative probabilities.

Example 

A call center receives on average 4 calls per hour. Use your calculator to find the probability of receiving at most 3 calls in one hour.
▶️ Answer/Explanation
We are looking for \( P(X \leq 3) \) where \( X \sim \text{Po}(4) \).

Using TI-Nspire:
• Press menu → 6: Probability → 5: Distributions → F: Poisson CDF
• Input: mean = 4, x = 3
• Result: \( \boxed{P(X \leq 3) \approx 0.4335} \)

Using Casio fx-CG50:
• Press MENU → STAT → DIST → P
• Choose Poisson → PoissonCD
• Input: Data: 3, λ = 4
• Result: \( \boxed{0.4335} \)

Relation Between Poisson and Exponential Distributions

If \( X \sim \text{Po}(m) \), the number of events in a fixed interval, then the time between successive events follows: $ T \sim \text{Exp}(\lambda = m) $

Limitations and Considerations

  • Events must occur independently.
  • Rate \( m \) must be constant over the interval.
  • Events must be discrete and countable.
  • For large \( m \), Poisson approximates Normal: $ \text{Po}(m) \approx \mathcal{N}(m, m) $
Scroll to Top