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Section 5 – Expectation and Other Distribution Parameters

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 Expectatio n: Discrete Continuous. Expectation of h(x). Discrete Continuous. Moments of a Random Variable.

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Section 5 – Expectation and Other Distribution Parameters

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  1. Section 5 – Expectation and Other Distribution Parameters

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

  3. Expectation of h(x) • Discrete • Continuous

  4. 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

  5. Variance of X • Notation • Definition:

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

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

  8. Properties of MGF’s

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

  10. Characteristics of a Distribution • Percentile: value of X, c, such that p% falls to the left of c • Median: p = .5, the 50th 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

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

  12. 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

  13. 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

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