Poisson distribution in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the Poisson distribution used for?

The Poisson distribution models the number of times an event happens in a fixed interval of time or space, when events occur independently and at a constant average rate.

beginner

What parameter does the Poisson distribution use?

It uses a single parameter called lambda (λ), which represents the average number of events in the interval.

intermediate

How do you calculate the probability of exactly k events in Poisson distribution?

The probability is given by P(k) = (λ^k * e^(-λ)) / k!, where k is the number of events, λ is the average rate, and e is Euler's number.

beginner

In Python's scipy library, which function generates Poisson probabilities?

The function is scipy.stats.poisson.pmf(k, mu), where k is the number of events and mu is the average rate λ.

beginner

What real-life example fits the Poisson distribution?

Counting the number of emails you receive in an hour, assuming emails come independently and at a steady average rate.

What does the parameter λ represent in a Poisson distribution?

A The variance of the events

B The average number of events in the interval

C The total number of events observed

D The time between events

λ (lambda) is the average number of events expected in the fixed interval.

Which scipy function calculates the probability mass function for Poisson?

A scipy.stats.poisson.pmf

B scipy.stats.poisson.cdf

C scipy.stats.norm.pdf

D scipy.stats.binom.pmf

pmf stands for probability mass function, which gives the probability of exactly k events.

If λ = 3, what is the probability of 0 events occurring?

A 3^0

B 0

C e^(-3)

D 1

P(0) = (3^0 * e^(-3)) / 0! = e^(-3).

Poisson distribution assumes events occur:

A Only once

B In clusters

C With increasing frequency over time

D Independently and at a constant average rate

Poisson assumes independent events with a steady average rate.

Which of these is NOT a typical use of Poisson distribution?

A Height of people in a group

B Number of cars passing a checkpoint

C Number of decay events from a radioactive source

D Number of phone calls received per hour

Height is continuous and usually modeled by normal distribution, not Poisson.

Explain the Poisson distribution and when it is used.

Think about counting rare events over time or space.

Describe how to calculate the probability of k events using the Poisson formula.

Remember the role of e and factorial in the formula.

Practice

1. What does the Poisson distribution model in data science?

easy

A. The probability of a variable being normally distributed

B. The average value of a continuous variable

C. The relationship between two categorical variables

D. The number of events happening in a fixed interval of time or space

Solution

1. Step 1: Understand the Poisson distribution concept

   The Poisson distribution is used to model counts of events that happen independently in a fixed interval of time or space.

2. Step 2: Match the concept to the options

   The number of events happening in a fixed interval of time or space correctly describes this as the number of events in a fixed interval, while others describe unrelated concepts.

3. Final Answer:

   The number of events happening in a fixed interval of time or space -> Option D

4. Quick Check:

   Poisson models event counts in intervals = A [OK]

Hint: Poisson counts events in fixed intervals, not averages or categories [OK]

Common Mistakes:

• Confusing Poisson with normal distribution
• Thinking Poisson models averages instead of counts
• Mixing Poisson with categorical data analysis

2. Which of the following is the correct way to import the Poisson distribution from scipy.stats?

easy

A. from scipy.stats import poisson

B. import poisson from scipy.stats

C. from scipy import poisson.stats

D. import scipy.poisson

Solution

1. Step 1: Recall correct Python import syntax

   Python imports use 'from module import object' format. The Poisson distribution is in scipy.stats as 'poisson'.

2. Step 2: Check each option's syntax

   from scipy.stats import poisson uses correct syntax. Options A, B, and C have incorrect import statements.

3. Final Answer:

   from scipy.stats import poisson -> Option A

4. Quick Check:

   Correct import syntax = D [OK]

Hint: Use 'from scipy.stats import poisson' to import Poisson [OK]

Common Mistakes:

• Using wrong import order or keywords
• Trying to import from wrong submodules
• Using 'import' instead of 'from ... import ...'

3. What is the output of this code?

from scipy.stats import poisson
rv = poisson(mu=3)
prob = rv.pmf(2)
print(round(prob, 3))

medium

A. 0.2240

B. 0.224

C. 0.149

D. 0.050

Solution

1. Step 1: Understand the pmf function with mu=3 and k=2

   The pmf gives the probability of exactly k=2 events when the average rate mu=3.

2. Step 2: Calculate the Poisson pmf for k=2, mu=3

   Using formula: P(k=2) = (3^2 * e^-3) / 2! ≈ 0.224

3. Final Answer:

   0.224 -> Option B

4. Quick Check:

   Poisson pmf(2,3) ≈ 0.224 [OK]

Hint: Use pmf(k) with mu to get exact event probability [OK]

Common Mistakes:

• Confusing pmf with cdf
• Rounding incorrectly
• Using wrong mu or k values

4. Identify the error in this code snippet:

from scipy.stats import poisson
rv = poisson(4)
prob = rv.pmf(3)
print(prob)

medium

A. poisson is not imported correctly

B. pmf cannot be called on rv object

C. No error, code runs correctly

D. The parameter should be named mu=4, not just 4

Solution

1. Step 1: Check the poisson object creation

   poisson(4) creates a frozen distribution with mean 4. The parameter can be passed positionally.

2. Step 2: Verify pmf method usage

   Calling pmf(3) on the frozen distribution is correct and prints the probability of 3 events.

3. Final Answer:

   No error, code runs correctly -> Option C

4. Quick Check:

   Frozen distribution with positional mu works = A [OK]

Hint: Frozen distribution accepts positional mu parameter [OK]

Common Mistakes:

• Thinking mu must be named explicitly
• Assuming pmf is not a method of frozen distribution
• Believing import is incorrect without error

5. You want to find the probability of observing at most 2 events when the average rate is 1.5. Which scipy code correctly calculates this?

hard

A. poisson.cdf(2, mu=1.5)

B. poisson.pmf(2, mu=1.5)

C. poisson.sf(2, mu=1.5)

D. poisson.cdf(1.5, mu=2)

Solution

1. Step 1: Understand the problem requirement

   We want the probability of at most 2 events, which means P(X ≤ 2).

2. Step 2: Identify the correct scipy function

   poisson.cdf(k, mu) gives P(X ≤ k). So poisson.cdf(2, mu=1.5) is correct.

3. Final Answer:

   poisson.cdf(2, mu=1.5) -> Option A

4. Quick Check:

   At most k events = cdf(k) [OK]

Hint: Use cdf(k) for probability of up to k events [OK]

Common Mistakes:

• Using pmf instead of cdf for cumulative probability
• Confusing survival function (sf) with cdf
• Swapping mu and k parameters