Lecture 1 slides: Distributions and normal random variables

Given a random variable X, we de ne a cumulative distribution function (cdf), FX : R → [0, 1], such that FX(t) = P{X ≤ t} for all t ∈ R. Here P{X ≤ t} denotes the probability that X ≤ t. To emphasize that random variable X has cdf FX , we write X ∼ FX . Note that FX(t) is a nondecreasing function of t. There are 3 types of random variables: discrete, continuous, and mixed. Discrete random variable, X, is characterized by a list of possible values, X = {x1, ..., xn}, and their probabilities, p = {p1, ..., pn}, where pi denotes the probability that X will take value xi, i.e. pi = P{X = xi} for all i = 1, ..., n. Note that p1 + ... + pn = 1 and pi ≥ 0 for all i = 1, ..., n by de nition of probability. Then the cdf of X is given by FX(t) = j=1,...,n: xj≤t pj . Continuous random variable, ∑