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.
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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
$P(X = 3) = \frac{e^{-2} \cdot 2^3}{3!} = 0.180$
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
$P(X=3 \text{ and } Y=4) = P(X=3) \cdot P(Y=4) = 0.1804 \cdot 0.0902 = 0.0163$
Sum of Independent Poisson Distributions
If:
$X \sim \text{Po}(m), \quad Y \sim \text{Po}(n), \quad X \text{ and } Y \text{ independent}$
Then:
$X + Y \sim \text{Po}(m + n)$
Example:
If call center A has $m = 2$, and B has $n = 3$:
▶️Answer/Explanation
$X + Y \sim \text{Po}(5), \quad P(X + Y = 6) = 0.161$
Cumulative Distribution Function (CDF)
$P(X \leq x), \quad P(X < x), \quad P(X \geq x)$
We use the Cumulative Distribution Function (CDF), accessible via GDC:
$\mathbf{Ped(x–y)}$: Calculates $P(x \leq X \leq y)$
Example:
$P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)$
▶️Answer/Explanation
$P(X < 3) = 0.135 + 0.271 + 0.271 = 0.677$
Relation Between Poisson and Exponential Distributions
$X \sim \text{Po}(m)$: Number of events in a fixed interval
Then: The time between successive events follows an Exponential distribution with rate $\lambda = m$:
$T \sim \text{Exp}(\lambda)$
This is the inter-arrival time of events.
Limitations and Considerations
Not appropriate if:
- Events do not occur independently.
- Rate $m$ varies over the interval.
- Events are not discrete or countable.
- For large values of $m$, Poisson approaches Normal distribution:
$\text{Po}(m) \approx \mathcal{N}(m, m)$
