Section 5 – Expectation and Other Distribution Parameters

Expected Value (mean) As the number of trials increases, the average outcome will tend towards E(X): the mean Expectation: – Discrete – Continuous

Expectation of h(x) Discrete Continuous

Moments of a Random Variable n: positive integer n-th moment of X: – So h(x) = X^n – Use E[h(X)] formula in previous slide n-th central moment of X (about the mean): – Not as important to know

Variance of X Notation Definition:

Important Terminology Standard Deviation of X: Coefficient of variation: – Trap: “Coefficient of variation” uses standard deviation not variance.

Moment Generating Function (MGF) Moment generating function of a random variable X: – Discrete: – Continuous:

Properties of MGF’s

Two Ways to Find Moments 1.E[e^(tx)] 2. Derivatives of the MGF

Characteristics of a Distribution Percentile: value of X, c, such that p% falls to the left of c – Median: p =.5, the 50 th percentile of the distribution (set CDF integral =.5) What if (in a discrete distribution) the median is between two numbers? Then technically any number between the two. We typically just take the average of the two though Mode: most common value of x – PMF p(x) or PDF f(x) is maximized at the mode Skewness: positive is skewed right / negative is skewed left – I’ve never seen the interpretation on test questions, but the formula might be covered to test central moments and variance at the same time

Expectation & Variance of Functions Expectation: constant terms out, coefficients out Variance: constant terms gone, coefficients out as squares

Mixture of Distributions Collection of RV’s X1, X2, …, Xk – With probability functions f1(x), f2(x), …, fk(x) – These functions have weights (alpha) that sum to 1 In a “mixture of distribution” these distributions are mixed together by their weights – It’s a weighted average of the other distributions

Parameters of Mixtures of Distributions Trap: The Variance is NOT a weighted average of the variances You need to find E[X^2]-(E[X)])^2 by finding each term for the mixture separately