Probability Distributions

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A random variable is a real valued function whose domain is the sample space of a random experiment. Example: Let X represent the number of heads when tossing a coin twice. X can take values 0, 1, or

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Discrete Random Variable: Takes distinct, countable values. Example: Number of students in a class (15, 16, 17, etc.). Continuous Random Variable: Takes infinite uncountable values in a range. Example

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1. Links every possible outcome with its probability 2. All probabilities are non-negative: P(xi) ≥ 0 3. Sum of all probabilities equals 1: Σpi = 1 4. Covers all elements of the sample space 5. Each p

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Mathematical Expectation (Expected Value) is the weighted average of all possible values of a random variable X. Formula: E(X) = Σ(xi × pi) = x₁p₁ + x₂p₂ + ... + xₙpₙ. It represents the theoretical me

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Sample space: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} X = number of heads: 0, 1, 2, 3 P(X=0) = 1/8, P(X=1) = 3/8, P(X=2) = 3/8, P(X=3) = 1/8 E(X) = 0×(1/8) + 1×(3/8) + 2×(3/8) + 3×(1/8) = 12/8 = 1.5

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Variance measures the average degree to which each value differs from the mean expectation. Formula: Var(X) = σ² = Σxi²pi - [Σxipi]² = E(X²) - [E(X)]². Importance: Shows variability/spread of data. Lo

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Bernoulli trials are independent trials with only two outcomes (success/failure). Four conditions: 1) Finite number of trials, 2) Trials are independent, 3) Each trial has exactly two outcomes (succes

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P(X = r) = ⁿCᵣ × pʳ × qⁿ⁻ʳ = n!/(r!(n-r)!) × pʳ × qⁿ⁻ʳ Where: n = number of trials, r = number of successes (0,1,2,...,n), p = probability of success in one trial, q = probability of failure = 1-p. Th