--- title: "Define random variables; properties" subject: "Simulation and Modeling" chapter: 6 key: "semester_8/simulation_and_modeling:rn-define-properties-variables" source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/rn-define-properties-variables site: BCT Engineering --- # Define random variables; properties Simulation and Modeling · Chapter 6: Random Number - Asked in: 2070 Magh, 2+3 marks - Video: [Random variables | Probability and Statistics | Khan Academy](https://www.youtube.com/watch?v=3v9w79NhsfI) — Khan Academy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/rn-define-properties-variables - Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-6 ## Question Define random variables. Explain properties of random variables. ## Short answer A random variable is a function mapping outcomes to real numbers; discrete with pmf p(x)=P(X=x), continuous with pdf f(x). Properties: p(x) in [0,1] (discrete), f(x) ≥ 0 (continuous); total probability sums/integrates to 1; CDF F(x)=P(X ≤ x) non‑decreasing; mean E(X) = Σ x p(x) or ∫ x f(x)dx, linear; variance V(X)=E(X²)−[E(X)]², V(aX+b)=a²V(X); if independent, E(XY)=E(X)E(Y) and V(X+Y)=V(X)+V(Y). ## Answer #### Random variable A random variable X is a function that assigns a real number to each outcome of a random experiment. Example: the number shown on a die, the inter-arrival time of customers, or the service time at a bank counter. - **Discrete** random variable: takes countable values (number of arrivals in an hour). Described by a probability mass function p(x) = P(X = x). - **Continuous** random variable: takes any value in an interval (service time). Described by a probability density function f(x), with P(a ≤ X ≤ b) = ∫ f(x) dx from a to b. #### Properties of random variables 1. **Probability bounds.** 0 ≤ p(x) ≤ 1 for discrete X, and f(x) ≥ 0 for continuous X. 2. **Total probability is 1.** Σ p(x) = 1 (discrete), ∫ f(x) dx = 1 over the whole range (continuous). 3. **Cumulative distribution function.** F(x) = P(X ≤ x). It is non-decreasing, with F(−∞) = 0 and F(+∞) = 1. 4. **Expected value (mean).** E(X) = Σ x·p(x) or ∫ x·f(x) dx. It is the long-run average. Also E(aX + b) = aE(X) + b, and E(X + Y) = E(X) + E(Y). 5. **Variance.** V(X) = E(X²) − [E(X)]². The standard deviation is σ = √V(X). Also V(aX + b) = a²V(X). 6. **Independence.** X and Y are independent if P(X = x, Y = y) = P(X = x)·P(Y = y). Then E(XY) = E(X)E(Y) and V(X + Y) = V(X) + V(Y). For example, the random number R uniform on [0, 1] has f(R) = 1, E(R) = 1/2 and V(R) = 1/12. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is a random variable? A. A fixed constant chosen at random B. A function that assigns a real number to each outcome of a random experiment C. Any number between 0 and 1 D. The sample mean Answer: B. A random variable maps each outcome of a random experiment to a real number. (Key point: Definition) 2. Which description of a continuous random variable is correct? A. Takes countable values and has a pmf B. Takes values in an interval and is described by a pdf with total area 1 C. Has probabilities that sum to 2 D. Always has f(x) <= 1 Answer: B. A continuous variable takes any value in an interval; f(x) >= 0 and its integral over the range is 1. (Key point: Discrete vs continuous) 3. What is the variance of a random variable and how does it scale? A. V(X) = E(X^2) - [E(X)]^2 and V(aX + b) = a^2 V(X) B. V(X) = E(X) and V(aX + b) = aV(X) + b C. V(X) = E(X^2) + [E(X)]^2 and V(aX+b) = a^2 V(X) + b D. V(X) = [E(X)]^2 and V(aX+b) = a V(X) Answer: A. V(X) = E(X^2) - [E(X)]^2; adding b does not change spread and scaling by a multiplies variance by a^2. (Key point: Variance) 4. If X and Y are independent, which holds? A. E(XY) = E(X) + E(Y) B. V(X + Y) = V(X) + V(Y) C. V(X + Y) = V(X) - V(Y) D. E(XY) = 0 Answer: B. For independent variables, E(XY) = E(X)E(Y) and V(X + Y) = V(X) + V(Y). 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