Module 3 · Distributions · Topic 1
Random Variable. A random variable is a numerical outcome of a random process. It assigns a number to each outcome in the sample space. Denoted by capital letters like X or Y.
Expected Value. The expected value E(X) is the long-run average of a random variable over many repetitions. It is the probability-weighted sum of all possible values.
E(X) = Σ[xᵢ × P(xᵢ)]
P(X = k) = C(n,k)p^k(1-p)^(n-k) — Exact probability of k successes in n independent Bernoulli trials.z = (x - mu) / sigma — Standardizes a value by measuring standard deviations from the mean.A game pays $0 (prob 0.5), $5 (prob 0.3), and $20 (prob 0.2). E(X) = 0(0.5) + 5(0.3) + 20(0.2) = 0 + 1.50 + 4.00 = $5.50.
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