1. Multivariate Probability Distributions

2. Multivariate Random Variables
In many settings, we are interested in 2 or more characteristics observed in experiments
Often used to study the relationship among characteristics and the prediction of one based on the other(s)
Three types of distributions:
Joint: Distribution of outcomes across all combinations of variables levels
Marginal: Distribution of outcomes for a single variable
Conditional: Distribution of outcomes for a single variable, given the level(s) of the other variable(s)

3. Joint Distribution

4. Marginal Distributions

5. Conditional Distributions
Describes the behavior of one variable, given level(s) of other variable(s)

6. Expectations

7. Expectations of Linear Functions

8. Variances of Linear Functions

9. Covariance of Two Linear Functions

10. Multinomial Distribution
Extension of Binomial Distribution to experiments where each trial can end in exactly one of k categories
n independent trials
Probability a trial results in category i is pi
Yi is the number of trials resulting in category I
p1+…+pk = 1
Y1+…+Yk = n

11. Multinomial Distribution

12. Multinomial Distribution

13. Conditional Expectations
When E[Y1|y2] is a function of y2, function is called the regression of Y1 on Y2

14. Unconditional and Conditional Mean

15. Unconditional and Conditional Variance

16. Compounding
Some situations in theory and in practice have a model where a parameter is a random variable
Defect Rate (P) varies from day to day, and we count the number of sampled defectives each day (Y)
Pi ~Beta(a,b) Yi |Pi ~Bin(n,Pi)
Numbers of customers arriving at store (A) varies from day to day, and we may measure the total sales (Y) each day
Ai ~ Poisson(l) Yi|Ai ~ Bin(Ai,p)