--- title: "Mean and variance of pdf 12x²(1−x)" subject: "Probability and Statistics" chapter: 3 key: "semester_5/probability_and_statistics:pdf-12x2-1-minus-x-mean" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/pdf-12x2-1-minus-x-mean site: BCT Engineering --- # Mean and variance of pdf 12x²(1−x) Probability and Statistics · Chapter 3: Continuous Probability Distributions - Asked in: 2076 Chaitra, 5 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/pdf-12x2-1-minus-x-mean - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3 ## Question Find the mean and variance of the probability density function given by f(x) = 12x²(1 − x); 0 ≤ x ≤ 1 0 ; otherwise ## Short answer For f(x)=12x²(1-x) on [0,1], mean E(X)=∫₀¹ x·12(x²−x³)dx=0.6; E(X²)=0.4; variance Var(X)=0.4−0.6²=0.04. ## Answer f(x) = 12x²(1 − x) = 12(x² − x³), 0 ≤ x ≤ 1. Check: ∫₀¹ 12(x² − x³) dx = 12(1/3 − 1/4) = 12 × 1/12 = 1, so it is a valid pdf. #### Mean E(X) = ∫₀¹ x · 12(x² − x³) dx = 12 ∫₀¹ (x³ − x⁴) dx = 12 (1/4 − 1/5) = 12 × 1/20 = **0.6** #### Variance E(X²) = ∫₀¹ x² · 12(x² − x³) dx = 12 ∫₀¹ (x⁴ − x⁵) dx = 12 (1/5 − 1/6) = 12 × 1/30 = 0.4 Var(X) = E(X²) − [E(X)]² = 0.4 − 0.36 = **0.04** **Answer: mean = 0.6, variance = 0.04 (standard deviation 0.2)** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A valid pdf requires: A. Integral of x f equals 1 B. Integral of f over its range equals 1 C. f(x) > 1 D. Mean equals 1 Answer: B. Total area is 1; here 12(1/3 - 1/4) = 1. (Key point: Validity) 2. E(X) is: A. Integral of f(x) dx = 1 B. Integral of x^2 f(x) dx = 0.4 C. 0.5 D. Integral of x f(x) dx = 0.6 Answer: D. 12(1/4 - 1/5) = 0.6. (Key point: Mean) 3. E(X^2) is: A. 12(1/5 - 1/6) = 0.4 B. 12(1/4 - 1/5) C. 0.36 D. 0.6 Answer: A. Integrate x^2 f(x). (Key point: E(X^2)) 4. Variance is: A. 0.4 + 0.36 B. 0.6 - 0.4 C. 0.4 - 0.36 = 0.04 D. 0.16 Answer: C. E(X^2) - mean^2. (Key point: Variance) ## Close variants set in other exams - [pdf Ax(6−x)²: find A, mean, variance](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-ax-6-minus-x-squared) — Probability and Statistics, chapter 3, asked 2× ## Related in Probability and Statistics - [pdf k(5+2x): find k, mean, variance](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-k-5-plus-2x-mean) — Probability and Statistics, chapter 3, asked 2× - [pdf kx³(4−x)²: k, mean, variance](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-kx3-4-minus-x-squared) — Probability and Statistics, chapter 3, asked 2× - [pdf Kx(2−x): K, mean, variance](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-kx-2-minus-x) — Probability and Statistics, chapter 3 - [Joint pdf a(x+y): marginals, probability](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Ajoint-pdf-a-x-plus-y) — Probability and Statistics, chapter 3 - [pdf x/2: library reserve time](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-x-over-2-reserve-time) — Probability and Statistics, chapter 3, asked 2× - [pdf kx: library reserve time, find k](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-kx-reserve-book-find-k) — Probability and Statistics, chapter 3, asked 2× ## Related in other subjects - [Fermi-Dirac function and 1.5kT probability](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Afermi-dirac-1-5kt) — Electrical Engineering Material, chapter 1 - [PERT expected time and completion probability](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_engineering_and_management%3Apert-probability-of-completion) — Industrial Engineering and Management, chapter 3