Tiêu đề 29 Mathematical Expectation.pdf ✅

MATHEMATICAL EXPECTATION 1. Definitions • Statistical Experiment – It is any process that generates several chance observations. • Sample Space – A collection or a set of possible outcomes of a random experiment. • Random Variable – A function that associates a real number with each element in the sample space • Discrete Sample Space – A sample space that contains a finite number of possibilities with as many elements as there are whole numbers • Continuous Sample Space – A sample space that contains an infinite number of possibilities equal to the number of points in a line segment 2. Expected Value For a discrete random variable X and its probability distribution f(x), the expected value is given by μ = E(x) = ∑x ⋅ f(x) On the other hand, for continuous random variables, μ = E(x) = ∫ x ⋅ f(x)dx ∞ −∞ 3. Variance of Random Variables For a discrete random variable, σ 2 = ∑(x − μ) 2 ⋅ f(x) For a continuous random variable, σ 2 = ∫ (x − μ) 2 ⋅ f(x)dx ∞ −∞