Mean, Variance, Moments and
Characteristic Functions
For a r.v X, its p.d.f f X (x ) represents complete information
about it, and for any set B on the x-axis
P X ( ) B f X ( x )dx.
(6-1)
B
Note that f X (x ) represents very detailed information, and
quit

3. Random Variables
Let (, F, P) be a probability model for an experiment,
and X a function that maps every , to a unique
point x R, the set of real numbers. Since the outcome
is not certain, so is the value X ( ) x. Thus if B is some
subset of R, we may

3. Random Variables
Let (, F, P) be a probability model for an experiment,
and X a function that maps every , to a unique
point x R, the set of real numbers. Since the outcome
is not certain, so is the value X ( ) x. Thus if B is some
subset of R, we may

Multiple Random Variables
Multiple Random Variables
Two Discrete Random Variables
Joint pmf
Marginal pmf
Two Continuous Random Variables
Joint Distribution (PDF)
Joint Density (pdf)
Marginal Densities
Independence
1
Multiple Random Variables
In many

6. Mean, Variance, Moments and
Characteristic Functions
For a r.v X, its p.d.f f X (x ) represents complete information
about it, and for any Borel set B on the x-axis
P X ( ) B f X ( x )dx.
(6-1)
B
Note that f X (x ) represents very detailed information,