MIT2_854F10_stats

# MIT2_854F10_stats - Statistical Inference Lecturer: Prof....

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1 Statistical Inference Lecturer: Prof. Duane S. Boning

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2 Agenda 1. 2. Sampling : Key distributions arising in sampling Chi-square, t, and F distributions 3. Estimation : Reasoning about the population based on a sample 4. Some basic confidence intervals Estimate of mean with variance known Estimate of mean with variance not known Estimate of variance 5. Hypothesis tests
3 Discrete Distribution: Bernoulli Bernoulli trial: an experiment with two outcomes Probability mass function (pmf): f(x) x ¾ p 1 - p ¼ 0 1

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4 Discrete Distribution: Binomial Repeated random Bernoulli trials n is the number of trials p is the probability of “success” on any one trial x is the number of successes in n trials
5 Binomial Distribution Binomial Distribution 0 0.05 0.1 0.15 0.2 0.25 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Number of "Successes" 0 0.2 0.4 0.6 0.8 1 1.2 Series1

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6 Discrete Distribution: Poisson Poisson is a good approximation to Binomial when n is large and p is small (< 0.1) Example applications: – # misprints on page(s) of a book – # transistors which fail on first day of operation Mean: Variance:
7 Poisson Distribution 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 Events per unit c Poisson Distribution 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

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8 Continuous Distributions • Uniform Distribution • Normal Distribution – Unit (Standard) Normal Distribution
9 Continuous Distribution: Uniform x a 1 b x a b cumulative distribution function * (cdf) probability density function (pdf) also sometimes called a probability distribution function *also sometimes called a cumulative density function

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10 Standard Questions You Should Be Able To Answer (For a Known cdf or pdf) x a 1 b x a b • Probability x sits within some range • Probability x less than or equal to some value
11 Continuous Distribution: Normal (Gaussian) 0 0.16 0.5 0.84 1 0.977 0.99865 0.0227 .00135 pdf cdf 0

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12 Continuous Distribution: Unit Normal Normalization cdf Mean Variance pdf
13 0.1 0.5 0.9 1 Using the Unit Normal pdf and cdf We often want to talk about “percentage points” of the distribution – portion in the tails

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14 Philosophy The field of statistics is about
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## This note was uploaded on 02/24/2012 for the course MECHANICAL 2.854 taught by Professor Stanleygershwin during the Fall '10 term at MIT.

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MIT2_854F10_stats - Statistical Inference Lecturer: Prof....

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