Probabilities, Statistics and Data Modeling

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Linear Regression

Simple Linear Regression

Overview of Simple Linear Regression

The parameters of the Least Squares Line

Coefficient of Determination R²

Geometric interpretation of Simple Linear Regression

Statistical properties of the parameters of SLR

• Bias, variance
• Covariance

Unbiased estimation of the measurement errors

Observations, residuals and predictions

The normality assumption. Confidence intervals

Validity test of a Simple Linear Regression

Leverage and influent observations

Multiple Linear Regression

Least Squares fitting of the model

Statistical properties of the estimated parameters

Residuals, adjusted values and errors

Coefficient of determination R², adjusted R²

The normality assumption

Confidence intervals

Test on nested models

Comparing MLR models

• Mallows' C_p, AIC, BIC

Variable selection in Multiple Linear Regression

Ridge regression

Definitions of Ridge Regression

Statistical properties of ridge estimators

Choosing the value of the ridge parameter

Ridge Regression and Principal Components Analysis

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Exponential, Gamma and Geometric distributions

Exponential distribution

Basic properties of the exponential distribution

• Mean, standard deviation, mode.
• Distribution function, quantiles.
• Moment generating function. Moments of all orders.

Maximum Likelihood estimation of the mean

Memoryless property

• Probabilistic aspect.
• Distributional aspect.

Strong memoryless property

Hazard rate function

First device to fail

Devices in series, parallel and stand-by

The "Law of diminishing returns", distribution of spacings

The "2// vs. 1" problem

Random sum of exponential random variables

Two problems (with solutions)

Gamma distribution

Basic properties

• Moment generating function.
• Mean, variance, mode.
• Additivity.

Geometric distribution

Basic properties

• Probability mass function.
• Cumulative distribution function.
• Direct calculation of the mean and variance.
• Moment generating function. Mean and variance revisited.
• Memoryless property.

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