Prove \( -1 \leq r(X, Y) \leq 1 \) for Karl Pearson’s Correlation Coefficient

Data Science and A.I. Lecture Series

Author: Bindeshwar Singh Kushwaha

Institute: PostNetwork Academy

Problem Statement

Prove that:

\[
-1 \leq r(X, Y) \leq 1
\]

The correlation coefficient \( r(X, Y) \) is a measure of the linear relationship between two variables \( X \) and \( Y \).

Formula for Correlation Coefficient

Step 1: Express the formula for \( r(X, Y) \)

The formula for \( r(X, Y) \) is:

\[
r(X, Y) = \frac{\sum_{i=1}^n (x_i – \overline{X})(y_i – \overline{Y})}{\sqrt{\sum_{i=1}^n (x_i – \overline{X})^2 \sum_{i=1}^n (y_i – \overline{Y})^2}}
\]

Let:

\[
x_i – \overline{X} = a_i \quad \text{and} \quad y_i – \overline{Y} = b_i
\]

Substituting:

\[
r(X, Y) = \frac{\sum_{i=1}^n a_i b_i}{\sqrt{\sum_{i=1}^n a_i^2 \sum_{i=1}^n b_i^2}}
\]

Proof using Cauchy-Schwarz Inequality

Step 2: Apply the Cauchy-Schwarz inequality

By the Cauchy-Schwarz inequality:

\[
\left( \sum_{i=1}^n a_i b_i \right)^2 \leq \left( \sum_{i=1}^n a_i^2 \right) \left( \sum_{i=1}^n b_i^2 \right)
\]

From this, we get:

\[
\left( \frac{\sum_{i=1}^n a_i b_i}{\sqrt{\sum_{i=1}^n a_i^2 \sum_{i=1}^n b_i^2}} \right)^2 \leq 1
\]

Thus:

\[
\left( r(X, Y) \right)^2 \leq 1
\]

Step 3: Conclude the proof

\[
-1 \leq r(X, Y) \leq 1
\]

PDF Presentation

corelprop2

Video

 

Reach PostNetwork Academy

• Website: PostNetwork Academy
• YouTube Channel: PostNetwork Academy YouTube
• Facebook Page: PostNetwork Academy Facebook
• LinkedIn Page: PostNetwork Academy LinkedIn

Thank You!