Probability Distributions

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A random variable is a real-valued function defined on the sample space of a random experiment. Domain: Sample space S, Co-domain: Real numbers R. Written as X : S → R. It assigns a unique real number

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Sample space S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} Possible values of X: {0, 1, 2, 3} [X = 0] = {TTT} [X = 1] = {HTT, THT, TTH} [X = 2] = {HHT, HTH, THH} [X = 3] = {HHH} This shows X is a discr

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Discrete Random Variable: • Finite or countably infinite values • Values obtained by counting • Examples: Number of students, dice outcomes Continuous Random Variable: • Uncountably infinite values f

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For discrete random variable X with possible values x₁, x₂, x₃, ... and probabilities p₁, p₂, p₃, ... The p.m.f. f(x) satisfies: 1. pᵢ ≥ 0 for all i 2. Σpᵢ = 1 3. pᵢ = P[X = xᵢ] = f(xᵢ) The p.m.f. g

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Sample space: 36 equally likely outcomes Possible values of X: {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} P(X=2) = 1/36, P(X=3) = 2/36, P(X=4) = 3/36 P(X=5) = 4/36, P(X=6) = 5/36, P(X=7) = 6/36 P(X=8) = 5/

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The c.d.f. F(x) of discrete random variable X is: F(x) = P[X ≤ x] = Σ(xᵢ≤x) P[X = xᵢ] Properties: • F(x) is non-decreasing • 0 ≤ F(x) ≤ 1 • F(-∞) = 0, F(∞) = 1 • F(x) is a step function for discrete

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Expected Value E(X) or μ is the weighted average of all possible values: E(X) = μ = Σᵢ₌₁ⁿ xᵢpᵢ = x₁p₁ + x₂p₂ + ... + xₙpₙ where xᵢ are possible values and pᵢ are corresponding probabilities. It rep

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Step 1: Identify possible values and probabilities X can take values: 1, 2, 3, 4, 5, 6 Each with probability: 1/6 Step 2: Apply formula E(X) = Σᵢ₌₁⁶ i × (1/6) E(X) = (1 + 2 + 3 + 4 + 5 + 6) × (1/6) E