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
Definition of a statistic
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
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