Probability density and cumulative functions in SciPy - Mini Project: Build & Apply

Probability Density and Cumulative Functions with SciPy

📖 Scenario: You work as a data analyst for a company that studies daily rainfall amounts. You want to understand the likelihood of different rainfall amounts using a common probability model called the normal distribution.

🎯 Goal: Build a small program that uses SciPy to calculate the probability density and cumulative probability for specific rainfall amounts using the normal distribution.

📋 What You'll Learn

Create a normal distribution with mean 50 and standard deviation 10

Calculate the probability density function (PDF) for rainfall amounts 40, 50, and 60

Calculate the cumulative distribution function (CDF) for rainfall amounts 40, 50, and 60

Print the PDF and CDF results clearly

💡 Why This Matters

🌍 Real World

Understanding rainfall probabilities helps in planning for agriculture, water management, and flood prevention.

💼 Career

Data analysts and scientists often use probability distributions to model and predict real-world events.

1

Create the rainfall amounts list

Create a list called rainfall_amounts with the exact values 40, 50, and 60.

SciPy

# Create the list rainfall_amounts with values 40, 50, and 60
# Your code here

Hint

Use square brackets to create a list with the numbers 40, 50, and 60 inside.

2

Set up the normal distribution parameters

Create two variables: mean set to 50 and std_dev set to 10.

SciPy

rainfall_amounts = [40, 50, 60]
# Create mean = 50 and std_dev = 10
# Your code here

Hint

Use simple assignment to create mean and std_dev variables with the given numbers.

3

Calculate PDF and CDF values using SciPy

Import norm from scipy.stats. Then create a normal distribution object called normal_dist using mean and std_dev. Use normal_dist.pdf() to calculate the PDF values for rainfall_amounts and store in pdf_values. Use normal_dist.cdf() to calculate the CDF values for rainfall_amounts and store in cdf_values.

SciPy

rainfall_amounts = [40, 50, 60]
mean = 50
std_dev = 10
# Import norm from scipy.stats
# Create normal_dist with mean and std_dev
# Calculate pdf_values for rainfall_amounts
# Calculate cdf_values for rainfall_amounts
# Your code here

Hint

Use norm(loc=mean, scale=std_dev) to create the distribution. Then call pdf() and cdf() with the list rainfall_amounts.

4

Print the PDF and CDF results

Print the pdf_values and cdf_values variables. Use print() statements with clear labels: "PDF values:" and "CDF values:".

SciPy

from scipy.stats import norm

rainfall_amounts = [40, 50, 60]
mean = 50
std_dev = 10

normal_dist = norm(loc=mean, scale=std_dev)
pdf_values = normal_dist.pdf(rainfall_amounts)
cdf_values = normal_dist.cdf(rainfall_amounts)
# Print PDF values with label
# Print CDF values with label
# Your code here

Hint

Use two print statements. Print the label first, then the variable.

Practice

1. What does the Probability Density Function (PDF) represent in a data distribution?

easy

A. The average value of the data

B. The likelihood of a value occurring at a specific point

C. The total probability up to a certain value

D. The maximum value in the data

Solution

1. Step 1: Understand PDF meaning

   The PDF shows how dense or likely values are at specific points in a distribution.

2. Step 2: Differentiate from CDF

   The CDF shows total probability up to a point, not the likelihood at a point.

3. Final Answer:

   The likelihood of a value occurring at a specific point -> Option B

4. Quick Check:

   PDF = Likelihood at point [OK]

Hint: PDF shows likelihood at exact points, not cumulative [OK]

Common Mistakes:

• Confusing PDF with CDF
• Thinking PDF gives total probability
• Assuming PDF gives average or max value

2. Which of the following is the correct way to import the cumulative distribution function (CDF) for a normal distribution from scipy.stats?

easy

A. import norm from scipy.stats; norm.cdf(x)

B. import scipy.stats.norm as norm; norm.cdf(x)

C. from scipy import norm; norm.cdf(x)

D. from scipy.stats import norm; norm.cdf(x)

Solution

1. Step 1: Recall scipy.stats import syntax

   The correct import for normal distribution functions is from scipy.stats import norm.

2. Step 2: Use norm.cdf(x) to get CDF

   After import, norm.cdf(x) computes the cumulative distribution function at x.

3. Final Answer:

   from scipy.stats import norm; norm.cdf(x) -> Option D

4. Quick Check:

   Correct import and usage = from scipy.stats import norm; norm.cdf(x) [OK]

Hint: Use 'from scipy.stats import norm' for normal distribution functions [OK]

Common Mistakes:

• Wrong import syntax causing errors
• Using 'import scipy.stats.norm as norm' which is invalid
• Trying to import norm directly from scipy

3. What is the output of the following code snippet?

from scipy.stats import norm
result = norm.pdf(0, loc=0, scale=1)
print(round(result, 3))

medium

A. 0.399

B. 0.500

C. 1.000

D. 0.000

Solution

1. Step 1: Understand norm.pdf at mean

   For a normal distribution with mean 0 and std 1, PDF at 0 is 1/sqrt(2*pi) ≈ 0.3989.

2. Step 2: Round the result

   Rounding 0.3989 to 3 decimals gives 0.399.

3. Final Answer:

   0.399 -> Option A

4. Quick Check:

   PDF at mean = 0.399 [OK]

Hint: PDF at mean of standard normal ≈ 0.399 [OK]

Common Mistakes:

• Confusing PDF with CDF value
• Rounding incorrectly
• Assuming PDF at mean is 0.5

4. Identify the error in this code that tries to calculate the CDF of a normal distribution at x=1:

import scipy.stats as stats
cdf_value = stats.norm.pdf(1, loc=0, scale=1)
print(cdf_value)

medium

A. Using pdf() instead of cdf() to get cumulative probability

B. Missing import statement for norm

C. Incorrect parameters for norm.pdf

D. Syntax error in print statement

Solution

1. Step 1: Check function used for CDF calculation

   The code uses norm.pdf which gives probability density, not cumulative probability.

2. Step 2: Correct function for cumulative probability

   To get cumulative probability, norm.cdf should be used instead of norm.pdf.

3. Final Answer:

   Using pdf() instead of cdf() to get cumulative probability -> Option A

4. Quick Check:

   Use cdf() for cumulative probability [OK]

Hint: Use cdf() for cumulative, pdf() for density [OK]

Common Mistakes:

• Mixing pdf and cdf functions
• Assuming pdf returns cumulative values
• Ignoring function purpose

5. You have a dataset of exam scores that roughly follow a normal distribution with mean 70 and standard deviation 10. Using scipy, how would you find the probability that a student scored less than 85?

hard

A. Use norm.cdf(70, loc=85, scale=10) to get cumulative probability

B. Use norm.pdf(85, loc=70, scale=10) to get cumulative probability

C. Use norm.cdf(85, loc=70, scale=10) to get cumulative probability

D. Use norm.pdf(70, loc=85, scale=10) to get cumulative probability

Solution

1. Step 1: Identify correct function for cumulative probability

   To find probability of scoring less than 85, use the cumulative distribution function norm.cdf.

2. Step 2: Use correct parameters for mean and std

   Set loc=70 (mean) and scale=10 (std deviation), and evaluate at 85.

3. Final Answer:

   Use norm.cdf(85, loc=70, scale=10) to get cumulative probability -> Option C

4. Quick Check:

   norm.cdf(value, loc=mean, scale=std) = cumulative probability [OK]

Hint: CDF with loc=mean, scale=std, value=threshold [OK]

Common Mistakes:

• Using pdf instead of cdf for cumulative probability
• Swapping mean and value parameters
• Confusing scale with loc