Module 3 · Distributions · Topic 3
What Is a Probability Distribution?. A probability distribution describes all possible values of a random variable and their associated probabilities. It completely specifies the behavior of the random variable.
Requirements. For a valid discrete distribution: every probability must be between 0 and 1, and all probabilities must sum to 1. For continuous distributions: the PDF must be non-negative, and the total area under the curve must equal 1.
Cumulative Distribution Function. The CDF, F(x) = P(X ≤ x), gives the probability that the random variable takes a value less than or equal to x. It is a non-decreasing function from 0 to 1.
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.Roll a die. X = face value. Distribution: P(1) = P(2) = ... = P(6) = 1/6. CDF: F(3) = P(X ≤ 3) = 3/6 = 0.5.
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