Chapter 5: Joint Probability

Distributions

Outline – Jointly Distributed

Random Variables

Outline – Expected values,

covariance and correlation

Outline – Statistics and their

distributions

Outline – Distribution of the

Sample Mean

Outline – The Distribution of

Linear Combinations

Joint Probability Mass Function

Joint Probability Density

Function

Example of joint probability

density

Marginal densities in example

Independent Random Variables

Conditional distributions

Expected value

Correlation

Definition of a statistic

Random samples

Simulation experiments

Steps in a simulation experiment

Simulating a sample mean from a

Weibull

Simulating a sample mean from a

Weibull (cont’d)

Characteristics of the simulated

values

Density plot and histogram

Normal probability plot of

sample means

Multiple sample sizes

Histograms of means of different

sizes of samples

Densities of means of different

sizes of samples

Example 5.23 – Simulating from

a skew distribution

Simulation

Histograms of means from a log-

normal distribution

Densities of means from a log-

normal distribution

Properties of sample mean and

sample sum

Central Limit Theorem

Convergence of means from U[-

1,1] to a normal shape

Histograms of raw means of

samples from U[-1,1].

Densities of scaled means of

samples from U[-1,1].

Linear Combinations and their

means

Variances of linear combinations

The difference between random

variables

The Case of Normal Random

Variables