--- title: "pdf kx²: lecture ending after bell" subject: "Probability and Statistics" chapter: 3 key: "semester_5/probability_and_statistics:pdf-kx2-lecture-bell" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/pdf-kx2-lecture-bell site: BCT Engineering --- # pdf kx²: lecture ending after bell Probability and Statistics · Chapter 3: Continuous Probability Distributions - Asked in: 2074 Asoj, 5 marks - Video: [Continuous Random Variable | PDF | Find k & Probabilities | Solved Problem](https://www.youtube.com/watch?v=Ui91zRVvlaA) — Mathematics Tutor - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/pdf-kx2-lecture-bell - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3 ## Question A college professor never finishes his lecture before the bell rings to end the period and always finishes his lectures within one minute after the bell rings. Let X = the time which elapses between the bell and the end of the lecture. Suppose that the p.d.f of X is f(x) = kx², 0 ≤ x ≤ 1 = 0, otherwise i) Find the value of k ii) What is the probability that the lecture ends within ½ minute of the bell ringing? iii) What is the probability that the lecture continues beyond the bell for between 15 and 30 seconds? ## Short answer Normalization gives k=3. P(X ≤ 0.5) = (0.5)^3 = 0.125. For 15–30 s (0.25–0.5 min), P = (0.5)^3 − (0.25)^3 = 0.109375 (7/64). ## Answer For a valid pdf, the total area under f(x) must be 1. #### i) Value of k ∫₀¹ kx² dx = k [x³/3]₀¹ = k/3 = 1, so **k = 3**. So f(x) = 3x², 0 ≤ x ≤ 1. #### ii) Lecture ends within ½ minute P(X ≤ ½) = ∫₀^½ 3x² dx = [x³]₀^½ = (½)³ = 1/8 = **0.125** #### iii) Lecture continues between 15 and 30 seconds 15 s = 0.25 minute and 30 s = 0.5 minute (X is in minutes). P(0.25 ≤ X ≤ 0.5) = [x³]₀.₂₅^₀.₅ = (0.5)³ − (0.25)³ = 0.125 − 0.015625 = **0.109375** (= 7/64) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is k determined for f(x) = kx^2 on 0 ≤ x ≤ 1? A. f(1) = 1 B. Total area ∫ f(x) dx = 1 C. Mean = 1 D. k = 1/2 by default Answer: B. A pdf must integrate to 1, so k/3 = 1 and k = 3. (Key point: Condition to find k) 2. P(X ≤ 1/2) with f(x) = 3x^2 equals: A. 1/2 B. 1/4 C. 1/8 D. 3/8 Answer: C. ∫0^(1/2) 3x^2 dx = (1/2)^3 = 1/8 = 0.125. (Key point: Probability as integral) 3. To find the chance the lecture runs between 15 and 30 seconds past the bell, what limits are used since X is in minutes? A. 15 and 30 B. 0.15 and 0.30 C. 0.25 and 0.5 D. 1/15 and 1/30 Answer: C. 15 s = 0.25 min and 30 s = 0.5 min; P = 0.125 − 0.015625 = 0.109375. (Key point: Unit conversion) ## Related in Probability and Statistics - [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× - [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 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 definition; f(x)=Kx(1−x) probabilities](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-kx-1-minus-x-probabilities) — Probability and Statistics, chapter 3 - [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 - [Corrected mean and SD after wrong entry](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Acorrected-mean-sd-20-students) — Probability and Statistics, chapter 1 ## Related in other subjects - [Phase diagrams before and after compaction](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aphase-diagram-compaction-consolidation) — Soil Mechanics, chapter 2 - [Sludge volume before and after digestion](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Asludge-volume-before-after-600kg) — Sanitary Engineering, chapter 8 - [Why DTFS uses only one period](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-vs-dtfs-duration) — Signals and Systems, chapter 2 - [Laplace: two switches, current before and after 8 ms](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Alaplace-two-switch-s1-s2-8ms) — Electrical Circuit Theory, chapter 4