Updated the questions with fixes

questions/184_empirical_probability_mass_function_(pmf)/meta.json

@@ -1,5 +1,5 @@
 {
-  "id": "182",
+  "id": "184",
   "title": "Empirical Probability Mass Function (PMF)",
   "difficulty": "easy",
   "category": "Probability & Statistics",
@@ -8,8 +8,8 @@
   "dislikes": "0",
   "contributor": [
     {
-      "profile_link": "https://github.com/jeetmukherjee",
-      "name": "jeetmukherjee"
+      "profile_link": "https://github.com/Jeet009",
+      "name": "Jeet Mukherjee"
     }
   ]
 }

questions/185_pmf_normalization_constant 2/learn.md

@@ -1,16 +1,41 @@
-## Solution Explanation
+## Learning: PMF normalization constant
 
-For a valid PMF, probabilities must sum to 1.
+### Idea and formula
+- **PMF requirement**: A probability mass function must satisfy ∑ p(xᵢ) = 1 and p(xᵢ) ≥ 0.
+- **Normalization by a constant**: If probabilities are given up to a constant, you determine that constant by enforcing the sum-to-1 constraint.
+  - If the form is p(xᵢ) = K · wᵢ with known nonnegative weights wᵢ, then
+    - ∑ p(xᵢ) = K · ∑ wᵢ = 1 ⇒ **K = 1 / ∑ wᵢ**.
+  - If the given expressions involve K in a more general way (e.g., both K and K² terms), still enforce ∑ p(xᵢ) = 1 and solve the resulting equation for K. Choose the solution that makes all probabilities nonnegative.
 
-Sum all terms:
+### Worked example (this question)
+Suppose the PMF entries are expressed in terms of K such that, when summed, the K-terms group as follows:
 
 - Linear in K: K + 2K + 2K + 3K + K = 9K
-- Quadratic in K: K^2 + 2K^2 + 7K^2 = 10K^2
+- Quadratic in K: K² + 2K² + 7K² = 10K²
 
-Therefore: 10K^2 + 9K = 1  =>  10K^2 + 9K - 1 = 0
+One concrete way to realize this is via the following table of outcomes and probabilities:
 
-Solve the quadratic: K = [-9 ± sqrt(81 + 40)] / 20 = [-9 ± 11] / 20
+| X  | p(X)         |
+|----|--------------|
+| x₁ | K + K²       |
+| x₂ | 2K + 2K²     |
+| x₃ | 2K           |
+| x₄ | 3K + 7K²     |
+| x₅ | K            |
 
-Feasible solution (K ≥ 0): K = 2/20 = 0.1
+These add up to 9K + 10K² as required.
 
-So the normalization constant is K = 0.1.
+Enforce the PMF constraint:
+
+- 9K + 10K² = 1 ⇒ 10K² + 9K − 1 = 0
+
+Quadratic formula reminder:
+
+- For aK² + bK + c = 0, the solutions are K = [−b ± √(b² − 4ac)] / (2a).
+
+Solve the quadratic:
+
+- K = [−9 ± √(9² + 4·10·1)] / (2·10) = [−9 ± √121] / 20 = [−9 ± 11] / 20
+- Feasible (K ≥ 0) root: K = (−9 + 11) / 20 = 2/20 = 0.1
+
+Therefore, the normalization constant is **K = 0.1**.

questions/185_pmf_normalization_constant 2/meta.json

@@ -1,15 +1,15 @@
 {
-  "id": "183",
+  "id": "185",
   "title": "Find PMF Normalization Constant",
-  "difficulty": "easy",
+  "difficulty": "medium",
   "category": "Probability & Statistics",
   "video": "",
   "likes": "0",
   "dislikes": "0",
   "contributor": [
     {
-      "profile_link": "https://github.com/jeetmukherjee",
-      "name": "jeetmukherjee"
+      "profile_link": "https://github.com/Jeet009",
+      "name": "Jeet Mukherjee"
     }
   ]
 }