Probabilistic Models for “Real-World” Processes
Many physical processes involve a random component – an element that cannot be described exactly by a deterministic algorithm.
A name for such a process is a “random experiment”. The term
“experiment” used here does not necessarily have its usual meaning
as a controlled situation in which treatments are randomly assigned
to units and reponses are observed, etc. Some examples of what we
consider to be “random experiments” are below. I’m sure you can
think of many more...
Examples of Random Experiments
• We try to access a web page and record the time it takes for the
webpage to respond. The time varies depending on how busy the
network is, the speed of the connection, and other factors beyond
our control.
• Consumers can send an email to an organization’s fraud box to
report a phishing attack. The organization records the number of
notifications and the time between notifications. Specific characteristics of the phishing attacks and the sensitivity of the victims
cause the number and the frequency of notifications to vary. A
regular pattern in the times of the notifications suggests that
the notifications are not genuine but that an attacker is sending emails to the fraud box to divert the organization’s attention
away from an attack.
• Dr. M. (one of the experimental design gurus in the statistics
department) wants to compare a new procedure to a standard
method for manufacturing computer chips, in hope that the new
procedure will lead to fewer defective chips. He randomly assigns 50 sets of parts to one procedure and 50 sets of parts to
the other procedure. He counts the number of defective chips
from both methods. (This is an example of an experiment in the
usual sense.) The number of defective chips is subject to sources
of variation besides differences between the two procedures. If
Dr. M. repeats the experiment, he may get a slightly different
outcome.
• A survey asks corn farmers how interested they are in selling
corn stover (residue from harvesting corn) to biorefineries for use
in biofuels. Each farmer indicates his/her interest level on a 1-5
scale, where a 1 indicates that the farmer is not interested at all
and a 5 means that the farmer is very interested. Even if we
identify factors that influence farmers’ interest levels, we can not
predict farmers’ responses exactly because individual preferences
that we have not identified cause farmers’ responses to vary.
Statistical analyses of probabilistic models that describe the random components of physical processes can help us better understand
these processes. One way (among many) to define probability and
statistics is as follows:
Probability - mathematical theory for modeling “experiments” where
outcomes occur randomly.
Statistics - theory of information that uses data to make inferences
about questions of interest, under the assumption that there is a random component to the process that generated the data.
Because statistical inference makes use of probability models, probability is a foundation for statistics.
Mathematical Models for Random Processes
To construct mathematically coherent probability models, we need
a formal framework for talking about random processes and the elements that comprise random experiments.
Definition: Probability Experiment - A process with random outcomes.
Examples:
1. A message can take two network routers to get to a recipient
computer. We record the status of router 1, the status of router
2, and the status of the recipient coputer, where the status is
either up (U) or down (D). We may also record if the message
was transmitted successfully (S) or if the transmission failed (F).
2. Record the time for a webpage to respond.
3. Toss a coin until we get a head.
Components of Probability Experiments
• Elementary Outcome (ω) - an outcome of a random process.
Examples
1. Network Routers:
ω = (router 1 down, router 2 down, recipient computer up)
= DDU
2. Time to access webpage: ω = 3.527 seconds
3. Toss a coin until a head: ω = T T T T H
• Sample Space (Ω) - set of all possible outcomes.
Examples
1. Network Routers:
Ω = {ordered triples of U’s and D’s}
= {DDD, DDU, DU D, U DD, U U D, U DU, DU U, U U U }
|Ω| = 8 = 23
2. Time to access webpage: Ω = (0, ∞)
3. Toss coin until a head: Ω = {H, T H, T T H, T T T H, . . .}
- Discrete Sample Space - sample space with a finite our countably infinite number of elements.
Examples
1. Network Routers: Discrete
2. Time to access webpage: Not discrete
3. Toss coin until a head: Discrete
- Note that there are usually multiple ways to express the sample space for a particular situation.
Example 3. Toss coin until a head: Ω = {1, 2, 3, . . .} is
an equivalent expression for the sample space
Ω = {H, T H, T T H, T T T H, . . .}.
• Event (A) - subset of Ω. (A collection of elementary outcomes).
Example
1. Suppose the message is transmitted successfully if at least one
router is up and the recipient computer is up.
A = Successful transmission
= {DDU, U DU, U U U }
2. Time to access a webpage:
B = More than 10 sectonds
= (10, ∞)
3. Toss coin until a head:
C = first head occurs between 5 and 11 tosses (inclusive)
= {T T T T H, T T T T T H, T T T T T T H, T T T T T T T H,
T T T T T T T T H, T T T T T T T T T H, T T T T T T T T T T H}
= {5, 6, 7, 8, 9, 10, 11}
Summary of the Components of a Probability
Experiement
Probability Experiments
Real World
Mathematical World
Observation/Experiment with random outcome
Random Experiment
List of all possible outcomes
Sample Space (Ω)
An individual outcome
Elementary Outcome (ω ∈ Ω)
Collection of outcomes
Event A ⊂ Ω