--- title: "pdf k(5+2x): find k, mean, variance" subject: "Probability and Statistics" chapter: 3 key: "semester_5/probability_and_statistics:pdf-k-5-plus-2x-mean" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/pdf-k-5-plus-2x-mean site: BCT Engineering --- # pdf k(5+2x): find k, mean, variance Probability and Statistics · Chapter 3: Continuous Probability Distributions - Asked in: 2076 Asoj, 5 marks - Video: [Random Variable - A continuous r.v x defined by f(x), to find k, mean and variable - P&S](https://www.youtube.com/watch?v=vk7xoL8odx8) — Arudra Maths Academy - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/pdf-k-5-plus-2x-mean - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3 ## Question If we the following probability density function f(x) = k(5 + 2x) for 2 ≤ x ≤ 4, 0 otherwise. Find the value of K and mean and variance of random variable X. ## Short answer k=1/22 from ∫₂⁴ k(5+2x)dx=22k=1. Mean=∫₂⁴ x(5+2x)dx/22=101/33≈3.0606; E(X²)=320/33≈9.6970, variance=9.6970−(101/33)²≈0.3297. ## Answer f(x) = k(5 + 2x), 2 ≤ x ≤ 4. #### Value of k ∫₂⁴ k(5 + 2x) dx = k [5x + x²]₂⁴ = k [(20 + 16) − (10 + 4)] = 22k = 1 **k = 1/22** #### Mean E(X) = (1/22) ∫₂⁴ (5x + 2x²) dx = (1/22) [5x²/2 + 2x³/3]₂⁴ ```text At x = 4: 5(16)/2 + 2(64)/3 = 40 + 42.667 = 82.667 At x = 2: 5(4)/2 + 2(8)/3 = 10 + 5.333 = 15.333 Difference = 67.333 ``` E(X) = 67.333/22 = 101/33 = **3.0606** #### Variance E(X²) = (1/22) ∫₂⁴ (5x² + 2x³) dx = (1/22) [5x³/3 + x⁴/2]₂⁴ ```text At x = 4: 5(64)/3 + 256/2 = 106.667 + 128 = 234.667 At x = 2: 5(8)/3 + 16/2 = 13.333 + 8 = 21.333 Difference = 213.333 ``` E(X²) = 213.333/22 = 320/33 = 9.6970 Var(X) = 9.6970 − (3.0606)² = 9.6970 − 9.3674 = **0.3297** **Answer: k = 1/22, mean = 3.0606, variance = 0.3297** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. k is found from: A. k(5+2x) = 1 B. Integral of k(5+2x) from 2 to 4 = 22k = 1 C. Integral from 0 to 4 D. Mean = 1 Answer: B. [5x + x^2] from 2 to 4 = 36 - 14 = 22. (Key point: Finding k) 2. k equals: A. 1/36 B. 22 C. 1/14 D. 1/22 Answer: D. 22k = 1. (Key point: Value of k) 3. E(X) is: A. Integral of x f(x) from 2 to 4 = 101/33 = 3.0606 B. Integral of f = 1 C. (2+4)/2 = 3 exactly D. Integral of x^2 f Answer: A. Weights x by the pdf; it is not the midpoint since f is not uniform. (Key point: Mean) 4. Variance is: A. 9.6970 B. 9.6970 + 9.3674 C. E(X^2) - mean^2 = 9.6970 - 9.3674 = 0.3297 D. 3.0606^2 Answer: C. Uses E(X^2) = 320/33. (Key point: Variance) ## Close variants set in other exams - [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× ## Related in Probability and Statistics - [Continuous random variable; pdf properties](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Acontinuous-rv-pdf-properties) — Probability and Statistics, chapter 3 - [Verify pdf 2e^(−2x); P(1