Poisson distribution in SciPy - Step-by-Step Execution

Concept Flow - Poisson distribution

Start with lambda (rate)

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Calculate PMF for k=0

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Calculate PMF for k=1

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Calculate PMF for k=n

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Output probabilities for each k

The Poisson distribution calculates the probability of a number of events (k) happening in a fixed interval, given the average rate (lambda).

Execution Sample

SciPy

from scipy.stats import poisson
lambda_ = 3
k_values = range(6)
pmf_values = [poisson.pmf(k, lambda_) for k in k_values]
print(pmf_values)

This code calculates the Poisson probabilities for k=0 to 5 events when the average rate is 3.

Execution Table

Step	k (events)	lambda (rate)	PMF Calculation	PMF Value
1	0	3	poisson.pmf(0, 3) = e^-3 * 3^0 / 0!	0.0498
2	1	3	poisson.pmf(1, 3) = e^-3 * 3^1 / 1!	0.1494
3	2	3	poisson.pmf(2, 3) = e^-3 * 3^2 / 2!	0.2240
4	3	3	poisson.pmf(3, 3) = e^-3 * 3^3 / 3!	0.2240
5	4	3	poisson.pmf(4, 3) = e^-3 * 3^4 / 4!	0.1680
6	5	3	poisson.pmf(5, 3) = e^-3 * 3^5 / 5!	0.1008
7	End	-	-	Finished calculating PMF for k=0 to 5

💡 All PMF values for k=0 to 5 calculated using lambda=3

Variable Tracker

Variable	Start	After 1	After 2	After 3	After 4	After 5	Final
k	N/A	0	1	2	3	4	5
pmf_values	[]	[0.0498]	[0.0498, 0.1494]	[0.0498, 0.1494, 0.2240]	[0.0498, 0.1494, 0.2240, 0.2240]	[0.0498, 0.1494, 0.2240, 0.2240, 0.1680]	[0.0498, 0.1494, 0.2240, 0.2240, 0.1680, 0.1008]

Key Moments - 2 Insights

Why does the PMF value for k=3 equal the value for k=2?

Because the Poisson formula depends on k and lambda, and for lambda=3, the probabilities for k=2 and k=3 happen to be very close, as shown in execution_table rows 3 and 4.

What does the lambda parameter represent in the Poisson distribution?

Lambda is the average number of events expected in the interval. It is fixed at 3 in this example, as shown in every row of the execution_table.

Visual Quiz - 3 Questions

Look at the execution_table, what is the PMF value when k=4?

A 0.2240

B 0.1680

C 0.1008

D 0.0498

💡 Hint

Check the row where k=4 in the execution_table under PMF Value column.

At which step does the PMF calculation for k=1 happen?

A Step 3

B Step 1

C Step 2

D Step 4

💡 Hint

Look at the Step column in execution_table for k=1.

If lambda was increased to 5, how would the PMF values for k=0 to 5 change?

A Values for higher k would increase, lower k decrease

B They would all decrease

C They would all increase

D Values for lower k would increase, higher k decrease

💡 Hint

Recall that lambda controls the average rate; higher lambda shifts probabilities toward higher k.

Concept Snapshot

Poisson distribution models count of events in fixed time.
Syntax: poisson.pmf(k, lambda)
Outputs probability of k events given average rate lambda.
PMF sums to 1 over all k.
Useful for rare event counts.
Lambda is mean and variance.

Full Transcript

The Poisson distribution calculates the chance of a number of events happening in a fixed time, based on an average rate called lambda. We calculate the probability for each number of events k using the formula poisson.pmf(k, lambda). In the example, lambda is 3, and we find probabilities for k from 0 to 5. The probabilities show how likely each count is. For instance, the chance of 0 events is about 5%, and for 3 events about 22%. Lambda controls the shape: higher lambda shifts probabilities to higher counts. This helps us understand event counts like calls per hour or arrivals per minute.