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

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Use PDF for likelihood

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Use CDF for cumulative probability

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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.