Workshop in Probability and Statistics

Continuing the discussion of basic probability we define complements ("not A") and examine how to find the probability of the complement of an event.

More on basic probability. How to find the probability of two or more events occurring when we use the terms "and" and "or." For instance, how to find the probability of events "A and B" / "A or B".

How to find the probability of multiple events all taking place when we know the probability of each event.

Introduction to conditional probability and how to solve using the fundamental probability equation.

Three examples of conditional probability questions solved.

How to calculate the intersection of several events. More examples using decision trees to calculate probabilities.

Bayes' Theorem and how to solve conditional probability questions using decision trees.

Putting it all together with Conditional Probability with a look ahead at Expected Value.

Definition and terms related to random variables and examples of probability distributions, including an explanation of cumulative probability.

Explanation and examples of expected value and its relationship to probability and statistics. Includes a refresher on weighted averages.

How to calculate the Expected Value and Standard Deviation of a function when it contains a Random Variable.

Graphing probability distributions in an X-Y coordinate plane. Calculating probabilities by measuring the area under a curve. Includes explanations of Histograms and the Uniform Distribution.

Introduction to the Normal Distribution and Z Scores. Explanation of how the number of standard deviations from the mean is related to probability.

Popular real estate website Wozill has developed an algorithm for predicting the eventual sales price of any house before it goes on the market.  Sometimes the estimate provided by the algorithm is high, and sometimes it is low, but overall the expected difference between the prediction given by the algorithm and the actual sales price of the home is zero--meaning that the averages of all predictions and recorded sales are the same.

Unfortunately, the standard deviation of the difference between the algorithm's predictions and the actual sales prices of the homes is rather large: $100k, normally distributed around $0.  Approximately what percentage of estimates provide by the Wozill algorithm will be $200k or more below the actual sales price of the home?

How Z Scores (# of standard deviations from the mean of a normal distribution) can be converted to cumulative probabilities. How to use the Standard Normal (Z) Table.

In this video we solve several problems related to probabilities and the Normal Distribution. Includes solving for observed values, expected values, standard deviations, and cumulative probabilities.

How to calculate confidence intervals using the Normal Distribution and Z Scores.

Definitions, examples, and how to calculate covariances and correlations for two random variables.

Portfolio Analysis has to do with how to calculate the joint variance (and standard deviation) of multiple random variables. This video includes the equation to calculate joint variances when there may be multiple instances of two random variable and the variables may be correlated.

An example illustrating the concepts of Portfolio Analysis as well as correlation and variance of Joint Random Variables.

Introduction to Sampling and the Central Limit Theorem. Also how the size of a sample relates to the accuracy of a prediction for a population parameter.

More on Sampling and the Central Limit Theorem. How to calculate the probability of observing a sample mean using the standard deviation of the sample.

How to apply the principles of Sampling and the Central Limit Theorem to proportions. Includes how to calculate a proportion sample standard deviation.

Definition of the t-distribution an how to perform sampling calculations when the standard deviation of the population is unknown. Also how to use the t-Table.

Several examples demonstrating calculations pertaining to Z values, sampling, confidence intervals, proportion sampling, and t-distributions. All related to the previous four videos: Stats 24-27.

Introduction to Hypothesis Testing and its relationship to Sampling. How to select null and alternative hypotheses and how to determine whether to use a one-tailed or two-tailed test.

Introduction to linear regression. Definitions of independent and dependent variables, scatterplots, best-fit lines, residuals, the least-squares method, and the prediction equation.

More on simple linear regression including how to analyze the output of regression analysis using example data. Definitions of R-squared, coefficients, and standard errors. Also how to test the significance of the relationship between an independent and dependent variable using hypothesis testing.

A grab bag of additional regression concepts including how to calculate confidence intervals for predicted changes to a dependent variable based on a change to an independent variable, degrees of freedom with multiple independent variables, standardized coefficients, and the F-statistic.

How to calculate confidence intervals for point predictions and population averages using regression.

Overview of the four main assumptions of linear regression: linearity, independence of errors, homoscedasticity, and normality of residual distribution.

Overview of multiple regression including the selection of predictor variables, multicollinearity, adjusted R-squared, and dummy variables.

Employing dummy variables and time-lagged variables to come up with a better predictive model for your multiple regression analysis.

This video provides a very brief overview of some ways that you can transform your data so that it takes the form of a linear function and can then be used in a regression. Includes exponential and logarithmic transformations.

An example illustrating the iterative process used to select predictor variables for a multiple regression model.

A quick introduction to ANOVA, including examples of one-way and two-way analysis of variance.