Random Variables and Probability Distributions

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A random variable is a function X : S → R that assigns a real number to each outcome in the sample space S of a random experiment. It transforms the outcomes of an experiment into numerical values. E

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Discrete Random Variable: Takes finite or countably infinite values. Example: Number of heads in coin tosses (0, 1, 2, 3, ...) Continuous Random Variable: Takes all real values in an interval. Exampl

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For a discrete random variable X with possible values x₁, x₂, x₃, ..., the probability distribution P(X = xᵢ) must satisfy: (i) P(xᵢ) ≥ 0 for every i (All probabilities are non-negative) (ii) Σ P(xᵢ)

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Sample Space S = {HH, HT, TH, TT} X can take values 0, 1, 2 Step 1: Find P(X = 0) = P(TT) = 1/4 Step 2: Find P(X = 1) = P(HT, TH) = 2/4 = 1/2 Step 3: Find P(X = 2) = P(HH) = 1/4 Probability Distribu

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For a discrete random variable X: Mean (μ) = Σ xᵢ P(X = xᵢ) Variance (σ²) = Σ (xᵢ - μ)² P(X = xᵢ) OR Variance (σ²) = Σ xᵢ² P(X = xᵢ) - μ² Standard Deviation (σ) = √(σ²) The second variance formula

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Step 1: Calculate Mean μ = Σ xᵢ P(X = xᵢ) μ = (-2)(1/8) + (-1)(2/8) + (0)(3/8) + (1)(1/8) + (2)(1/8) μ = (-2 - 2 + 0 + 1 + 2)/8 = -1/8 Step 2: Calculate Variance using σ² = Σ xᵢ² P(X = xᵢ) - μ² Σ xᵢ²

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Sample Space: {1, 2, 3, 4, 5, 6}, each with probability 1/6 Step 1: Calculate Mean μ = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) μ = (1+2+3+4+5+6)/6 = 21/6 = 7/2 Step 2: Calculate Variance

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A discrete random variable X follows binomial distribution with parameters n and p if: P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ, where x ∈ {0, 1, 2, ..., n} Where: • n = number of independent trials • p = probability