Random Variables (RVs) Motivation
The outcome of a random experiment need not be a
number (coin flip, horse race, etc.)
Usually interested in some number-valued
measurement of an outcome, rather than the
outcome itself (unless the outcomes are numbers alr
Estimation (Y&G Ch 9 in 2nd ed. or Ch. 12 in 3rd ed. )
Estimation refers to a systematic procedure
for evaluating some hidden parameter or
RV realisation , based on a realisation x of a
related RV X.
Well assume we know the conditional CDF
of X given =.
Probability and Random Models
Semester 1, 2014
and Tutorial Sheets
Based on earlier lecture notes by Girish Nair and Brian Krongold
PaRM is an introduction to probability, random variables, estimation, and stochas
(Y & G sec.s 10.1 10.4, 10.8 10.12)
Recall that a random variable is a mapping from the sample space to
the set of reals. Similarly, a random n-vector is a mapping from to ndim. real space.
Analogously, a stochastic process (= random
(sec. 8.2-8.3, Y&G)
Say we must make a decision about the possible values of some
unknown parameter or RV sample, , based on available data.
This is hidden, i.e. it is not observed directly. Instead, a sample
of a related RV X is obs
Sums of Random Variables
Many macroscopic events are due to accumulated effects of
a large # of microscopic events. Some examples include:
Gas Pressure: Many collisions between molecules and container
Interest Rates: Financial transactions of a lar
Back to stochastic processes.
Recall the Poisson random variable
from earlier lectures, see next slide
N is a Poisson RV with parameter if
is the average # of event occurrences in a
specified time interval, say [0, t], where t is fix
Multiple Random Variables
We want to study the joint probabilistic behavior of
multiple RVs defined on the same sample space
In real-life applications, different RVs correspond
to related physical parameters; their joint behavior
is thus often of great in
Outcome (): value or result of an observation.
Sample space (S/): is the set of all possible outcomes of a given experiment. Each
outcome is distinct.
Event (E): a collection of outcomes (part (can be whole or none) of the sample spac