Poisson distribution in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Poisson PMF Calculation

What is the output of this code that calculates the Poisson probability mass function (PMF) for k=3 events with an average rate (lambda) of 2?

SciPy

from scipy.stats import poisson
result = poisson.pmf(3, 2)
print(round(result, 4))

A 0.1804

B 0.2707

C 0.1353

D 0.2240

💡 Hint

Recall the Poisson PMF formula: P(k; λ) = (λ^k * e^-λ) / k!

The Poisson PMF for k=3 and λ=2 is (2^3 * e^-2) / 3! = 8 * e^-2 / 6 ≈ 0.1804.

❓ data_output

intermediate

Number of Events with High Probability

Using a Poisson distribution with λ=4, how many events (k) have a probability mass function (PMF) value greater than 0.1? Use k values from 0 to 10.

SciPy

from scipy.stats import poisson
pmf_values = [poisson.pmf(k, 4) for k in range(11)]
count = sum(p > 0.1 for p in pmf_values)
print(count)

A 5

B 4

C 6

D 7

💡 Hint

Calculate PMF for k=0 to 10 and count how many exceed 0.1.

PMF values for k=2 to 6 exceed 0.1, so the count is 5.

❓ visualization

advanced

Plotting Poisson PMF for Different Lambda Values

Which option produces a plot showing Poisson PMF curves for λ=2, 4, and 6 over k=0 to 10?

SciPy

import matplotlib.pyplot as plt
from scipy.stats import poisson
import numpy as np
k = np.arange(0, 11)
lambdas = [2, 4, 6]
for lam in lambdas:
    plt.plot(k, poisson.pmf(k, lam), marker='o', label=f'λ={lam}')
plt.xlabel('Number of events (k)')
plt.ylabel('PMF')
plt.title('Poisson PMF for different λ')
plt.legend()
plt.grid(True)
plt.show()

A Plots a single curve with λ=4 only, ignoring others.

B Plots three identical flat lines at y=1 for all k values.

C Plots three curves with peaks shifting right as λ increases, all between 0 and 1 on y-axis.

D Raises a NameError due to missing import of numpy.

💡 Hint

Check if all λ values are plotted and curves shift right with increasing λ.

The correct plot shows three distinct PMF curves for λ=2, 4, and 6, each peaking near their λ value.

🧠 Conceptual

advanced

Understanding Poisson Distribution Assumptions

Which statement correctly describes a key assumption of the Poisson distribution?

A The distribution models continuous data with normal shape.

B Events occur in fixed intervals with increasing probability over time.

C Events are dependent and occur in clusters.

D Events occur independently and the average rate is constant over time.

💡 Hint

Think about how events happen in a Poisson process.

Poisson assumes events happen independently and at a constant average rate.

🔧 Debug

expert

Identify the Error in Poisson PMF Calculation Code

What error does this code produce when calculating Poisson PMF for k=3 and λ='2' (string instead of number)?

SciPy

from scipy.stats import poisson
result = poisson.pmf(3, '2')
print(result)

A No error, prints a valid probability

B TypeError: unsupported operand type(s) for ** or pow(): 'str' and 'int'

C ValueError: lambda must be positive

D SyntaxError: invalid syntax

💡 Hint

Check the data type of lambda parameter and how pow() behaves with strings.

Passing a string instead of a number causes a TypeError during power calculation inside PMF.

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