--- title: "Continuous random variable; pdf properties" subject: "Probability and Statistics" chapter: 3 key: "semester_5/probability_and_statistics:continuous-rv-pdf-properties" source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/continuous-rv-pdf-properties site: BCT Engineering --- # Continuous random variable; pdf properties Probability and Statistics · Chapter 3: Continuous Probability Distributions - Asked in: 2073 Shrawan, 5 marks - Video: [An Introduction to Continuous Probability Distributions](https://www.youtube.com/watch?v=OWSOhpS00_s) — jbstatistics - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/continuous-rv-pdf-properties - Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3 ## Question Define continuous random variable with suitable example. Describe the properties of probability density function and distribution function. ## Short answer A continuous r.v. takes uncountably many values in an interval (e.g., lifetime in hours). Its pdf f(x)≥0 and integrates to 1; probabilities are areas. The cdf F(x)=∫ f(t) dt is non‑decreasing, limits 0 and 1, and f(x)=F'(x). Example: f(x)=2x on [0,1] gives F(x)=x^2. ## Answer **Continuous random variable:** a random variable that can take any value in an interval (or in the whole real line), so its possible values are uncountable. Its probabilities are found as areas under a curve, not from individual values; P(X = a) = 0 for every single value *a*. **Example:** the time taken (in minutes) by a student to load a web page, the life of a bulb in hours, or the height of a student. If X = life of a bulb, X can be 1000.5 h, 1000.57 h, and so on. #### Probability density function (pdf) A function f(x) used to describe a continuous random variable, so that P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. **Properties of pdf** 1. f(x) ≥ 0 for all x (a density is never negative). 2. Total area under the curve is one: ∫₋∞^∞ f(x) dx = 1. 3. P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx = area between *a* and *b*. 4. P(X = a) = ∫ₐᵃ f(x) dx = 0, so P(a < X < b) = P(a ≤ X ≤ b) = P(a ≤ X < b) = P(a < X ≤ b). 5. f(x) itself is not a probability and may exceed 1; only the area is a probability. #### Distribution function (cdf) F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt **Properties of distribution function** 1. 0 ≤ F(x) ≤ 1 for all x. 2. F(x) is non-decreasing: if a < b then F(a) ≤ F(b). 3. F(−∞) = 0 and F(+∞) = 1. 4. F(x) is continuous (for continuous X) and P(a < X ≤ b) = F(b) − F(a). 5. f(x) = dF(x)/dx wherever the derivative exists. **Example:** for f(x) = 2x, 0 ≤ x ≤ 1: F(x) = x², and P(0.5 < X < 1) = F(1) − F(0.5) = 1 − 0.25 = 0.75. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For a continuous random variable X and any single value a, P(X = a) is: A. f(a) B. 0 C. 1/2 D. F(a) Answer: B. Probabilities are areas under the curve, and a single point has zero area. (Key point: Continuous variable) 2. Which statement about a pdf f(x) is correct? A. f(x) cannot exceed 1 B. f(x) can exceed 1; only areas are probabilities C. f(x) can be negative D. f(x) always equals P(X = x) Answer: B. f(x) is a density; only the area under it is a probability, so f(x) may exceed 1 but never be negative. (Key point: Density is not probability) 3. Which is a property of the distribution function F(x)? A. F is non-decreasing with F(−∞) = 0 and F(∞) = 1 B. F(x) can decrease C. F(∞) = 0 D. F(x) can exceed 1 Answer: A. F(x) = P(X ≤ x) rises from 0 to 1 and never decreases. (Key point: Properties of cdf) 4. How is the pdf obtained from the cdf? A. f(x) = ∫F(x) dx B. f(x) = dF(x)/dx C. f(x) = 1 − F(x) D. f(x) = F(x)^2 Answer: B. The pdf is the derivative of F wherever it exists; e.g. F = x^2 gives f = 2x. 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