QM 10005

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Intro to QM

- Order statistic

- Mean, Maximum, Minimum, Range

- Median

- Mode

- Frequency distribution & Relative frequency distribution

- Histogram and distribution shapes

- Time series plots

- Percentiles and quartiles

- Measures of variability (variance, SD, IQR, CV)

- Scatter plots

- Covariance and correlation

Stats and Probabilities

- Scatter plots, correlation and interpretation

- Random experiment, outcomes events, sample space

- Probability and its properties

- Complements of events

- Union, intersection

- Joint, marginal, conditional probabilities

- Addition and multiplication rules

- Independence

- Bayes theorem

- Law of total probability

- Translating words into events and probabilities

Probability distribution

- Random variables

- Probability distribution

- Expected value

- Addition rule for expected values

- Variance and standard deviation

- Joint and marginal probability distribution

- Covariance and correlation with random variables

- Addition rule for variances

Conditional Expectations, Continuous Random Variables and The Normal Distribution

- Conditional probability with random variables

- Conditional expected values

- Law of Iterated Expectations

- Conditional variance

- Continuous vs Discrete random variable

- Probability Density Function

- Uniform Distribution

- Normal Distribution

- Expected value, variance, and rules

- Probabilities for the normal distribution

- Quantiles for the normal distribution

The T and Pareto distribution + Statistical Inference

- The t distribution and comparison to the normal

- Moments- mean, variable, skewness, kurtosis

- The Pareto distribution

- Probability calculations

- Concept of statistical inference

- Simple random sample

- Estimation of a mean

- Estimation of proportion

Sampling Distribution, Unbiasedness and CLT

- Concept of sampling distribution

- Unbiased estimator

- The sample mean = unbiased estimator of the population mean

- Variance of the sample mean

- The CLT

Hypothesis Testing + Hypothesis Testing for Mean

- The t distribution

- Hypothesis null and alternative

- Decision rules (p value, t stats) for one tailed & two tailed tests

- Hypothesis testing for a mean

- T statistic

- Significance level

- Critical value decision rules

- p-value decision rules

Hypothesis testing error + Confidence Intervals

- Type I error

- Type II error + Power

- Definition of a Confidence Interval for a mean

- Relationship between confidence interval and hypothesis testing

- Meaning of 95% confidence level and its connection to 5% s.f level

- Calculation and interpretation from a sample (including proportion)

Comparing Two Means + Inference

- Matched pair vs Independent samples

- Hypothesis tests and C.I using these sampling distributions

- Application of these

Linear Regression

- The linear regression model

- Least squares estimation

- Estimate a conditional mean for a given value of X

- Population and Sample Linear Regression Models

- Least square regression estimation

- Calculate sample conditional mean from sample regression

- Fitted values, Residuals

- Goodness of fit, R^2

Statistical Inference with Regression

- Interpretation of regression coefficient

- Presentation of regressions

- Hypothesis testing for regression coefficient

- Confidence interval for regression coefficient

- Statistical inference for regressions coefficient

- Confidence Interval for conditional means

- Connection between regression and correlation

- Problem: Outliers

- Problem: Heteroskedasticity

- Problem: Time series data