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Author: hardik1408

Problems/117_random_forest/learn.md

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+# Learn: Gini Impurity and Best Split in Decision Trees
+
+## Overview
+
+A core concept in Decision Trees (and by extension, Random Forests) is how the model chooses where to split the data at each node. One popular criterion used for splitting is **Gini Impurity**.
+
+In this task, you will implement:
+- Gini impurity computation
+- Finding the best feature and threshold to split on based on impurity reduction
+
+This helps build the foundation for how trees grow in a Random Forest.
+
+---
+
+## Gini Impurity
+
+For a set of samples with class labels \( y \), the Gini Impurity is defined as:
+
+$$
+G(y) = 1 - \sum_{i=1}^{k} p_i^2
+$$
+
+Where \( p_i \) is the proportion of samples belonging to class \( i \).
+
+A pure node (all one class) has \( G = 0 \), and higher values indicate more class diversity.
+
+---
+
+## Gini Gain for a Split
+
+Given a feature and a threshold to split the dataset into left and right subsets:
+
+$$
+G_{\text{split}} = \frac{n_{\text{left}}}{n} G(y_{\text{left}}) + \frac{n_{\text{right}}}{n} G(y_{\text{right}})
+$$
+
+We choose the split that **minimizes** $( G_{\text{split}} )$.
+
+---
+
+## Problem Statement
+
+You are given a dataset $( X \in \mathbb{R}^{n \times d} )$ and labels $( y \in \{0, 1\}^n $). Implement the following functions:
+
+### Functions to Implement
+
+```python
+def find_best_split(X: np.ndarray, y: np.ndarray) -> Tuple[int, float]:
+    ...
+```