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.