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Understanding Distribution Parameters and Expectation in Statistics

Learn about expectation, moments, variance, and other distribution parameters in statistical analysis. Explore MGF, characteristics of distributions, and functions. Understand mixed distributions and their key properties.

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Understanding Distribution Parameters and Expectation in Statistics

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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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