Example: LR

Logistic Regression — Mathematical Example

Problem Statement

Predict whether a student will pass or fail based on study hours.

0 = Fail
1 = Pass

Dataset

Step 1: Logistic Regression Formula

Logistic Regression uses the sigmoid function:

P(y = 1) = 1 / (1 + e^(-z))

Where:

z = b0 + b1x

Step 2: Assume Model Values

Suppose the model learned:

b0 = -6
b1 = 2

So:

z = -6 + 2x

Step 3: Predict for a New Student

Suppose:

Study Hours = 4

Substitute x = 4:

z = -6 + 2(4)

z = -6 + 8

z = 2

Step 4: Apply Sigmoid Function

P(y = 1) = 1 / (1 + e^(-2))

Approximate value:

e^(-2) = 0.135

So:

P(y = 1) = 1 / (1 + 0.135)

P(y = 1) = 1 / 1.135

P(y = 1) = 0.881

Step 5: Apply Decision Rule

Usually, threshold is:

0.5

Decision rule:

If probability >= 0.5 → Predict 1
If probability < 0.5 → Predict 0

Here:

0.881 >= 0.5

So prediction is:

1

Final Prediction

The student is predicted to PASS.

Another Example

Predict for:

Study Hours = 2

Calculate z:

z = -6 + 2(2)

z = -6 + 4

z = -2

Apply sigmoid:

P(y = 1) = 1 / (1 + e^(-(-2)))

P(y = 1) = 1 / (1 + e^2)

Approximate:

e^2 = 7.389

P(y = 1) = 1 / (1 + 7.389)

P(y = 1) = 1 / 8.389

P(y = 1) = 0.119

Since:

0.119 < 0.5

Prediction is:

0

Final Prediction

The student is predicted to FAIL.

Summary

In Logistic Regression, the model first calculates a linear value z. Then the sigmoid function converts this value into a probability between 0 and 1. Finally, a threshold value, usually 0.5, is used to classify the output as class 0 or class 1.