Probability density and cumulative functions in SciPy - Step-by-Step Execution

Concept Flow - Probability density and cumulative functions

Start with a distribution

↓

Calculate PDF at x

↓

Calculate CDF at x

↓

Use PDF for likelihood

↓

Use CDF for cumulative probability

↓

End

Start with a probability distribution, then calculate the PDF and CDF values at a point x to understand likelihood and cumulative probability.

Execution Sample

SciPy

from scipy.stats import norm
x = 0
pdf_val = norm.pdf(x)
cdf_val = norm.cdf(x)
print(pdf_val, cdf_val)

Calculate the PDF and CDF of the standard normal distribution at x=0.

Execution Table

Step	Action	Input x	PDF Calculation	PDF Result	CDF Calculation	CDF Result
1	Import norm from scipy.stats	-	-	-	-	-
2	Set x value	0	-	-	-	-
3	Calculate PDF at x	0	norm.pdf(0)	0.3989422804014327	-	-
4	Calculate CDF at x	0	-	-	norm.cdf(0)	0.5
5	Print results	-	0.3989422804014327	0.3989422804014327	0.5	0.5
6	End	-	-	-	-	-

💡 Finished calculating PDF and CDF for x=0 in standard normal distribution.

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	Final
x	undefined	0	0	0	0
pdf_val	undefined	undefined	0.3989422804014327	0.3989422804014327	0.3989422804014327
cdf_val	undefined	undefined	undefined	0.5	0.5

Key Moments - 2 Insights

Why is the PDF value at x=0 less than 1 even though it looks like a probability?

PDF values are densities, not probabilities. They can be less than 1 because they represent height under the curve, not direct probabilities. See execution_table row 3.

Why is the CDF value at x=0 exactly 0.5?

CDF gives cumulative probability up to x. For standard normal, half the data is below 0, so CDF at 0 is 0.5. See execution_table row 4.

Visual Quiz - 3 Questions

Look at the execution table, what is the PDF value at step 3?

A 0.5

B 0.3989

C 1.0

D 0.0

💡 Hint

Check the 'PDF Result' column in row 3 of the execution_table.

At which step does the CDF get calculated?

A Step 4

B Step 3

C Step 2

D Step 5

💡 Hint

Look at the 'Action' column for 'Calculate CDF at x' in the execution_table.

If x was changed to 1, how would the PDF value change compared to step 3?

A It would increase

B It would stay the same

C It would decrease

D It would become zero

💡 Hint

Recall that PDF of normal distribution peaks at 0 and decreases as x moves away from 0.

Concept Snapshot

Probability Density Function (PDF) gives the height of the curve at x, showing likelihood density.
Cumulative Distribution Function (CDF) gives total probability up to x.
Use scipy.stats.norm.pdf(x) and .cdf(x) for standard normal.
PDF values can be less than 1; CDF values range from 0 to 1.
PDF is for density, CDF is for cumulative probability.

Full Transcript

We start with a probability distribution, here the standard normal. We pick a point x=0. We calculate the PDF at x=0 using norm.pdf, which gives about 0.3989. This is the height of the curve at zero, showing likelihood density. Then we calculate the CDF at x=0 using norm.cdf, which gives 0.5, meaning half the data lies below zero. We track variables x, pdf_val, and cdf_val through each step. Beginners often confuse PDF values with probabilities; PDF is a density, not a direct probability. Also, CDF at zero is 0.5 because the normal distribution is symmetric. Changing x affects PDF and CDF values accordingly.

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