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Stat 330 (Spring 2015): slide set 1
Because statistical inference makes use of probability models, probability
is a foundation for statistics. To use probability and statistics in a
mathematically coherent way, we need a formal framework for talking about
random processes and the elements that comprise random experiments.
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.
Probability - mathematical theory for modeling experiments where
outcomes occur randomly.
We want to study physical processes that are not completely deterministic.
Using probability and statistics to understand the random components of
such processes can help us do this. Two definitions (among many that I’ve
seen) of probability and statistics are as follows.
Probability and Statistics
Last update: January 9, 2015
Stat 330 (Spring 2015)
Slide set 1
Some examples of what we consider to be random experiments are
below. More interesting examples can be found in the textbook or Prof.
Hofmann’s notes.
The term experiment used here does not necessarily have its usual
meaning of a controlled situations in which outputs (responses) are
observed as a result of inputs (factors).
A name for such a process is a random experiment.
Many physical processes involve a random component – an element that
cannot be described exactly by a deterministic algorithm.
Probabilistic Models for “Real-World” Processes
Stat 330 (Spring 2015): slide set 1
Uncertainty appears in all areas of computer science and engineering.
We will see some realistic examples as we proceed.
Uncertainty exists in many aspects of science, business and our everyday
life. It is something usually unavoidable which we have to deal with.
The lack of certainty, a state of having limited knowledge where it is
impossible to exactly describe current state or future outcome, (or the
existence of more than one possible outcome).
Stat 330 (Spring 2015): slide set 1
Stat 330 (Spring 2015): slide set 1
A company measures the installation time of a software system under
different conditions so that it can give customers some idea of the time
Consumers can send an email to an organization’s phishing box to report
a phishing attempt. The organization records the number of notifications
and the time between notifications.
Components of Random Experiments (Continued)
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 }
and |Ω| = 8 = 23
Ω = (0, ∞)
3. Network Routers:
Ω = {H, T H, T T H, T T T H, . . .}
2. Time to access webpage:
1. Toss coin until a head:
Sample Space (Ω) - set of all possible outcomes.
ω = (router 1 down, router 2 down, recipient computer up) = DDU
ω = 3.527 seconds
3. A message can take two network routers to get to a recipient computer.
We may record the status of router 1, the status of router 2, and the
status of the recipient computer, where the status is either up (U) or
down (D).
Example 1. Toss coin until a head: Ω = {1, 2, 3, . . .} is an equivalent
expression for the sample space.
– Note that there are usually multiple ways to express the sample space
for a particular experiment.
1. Toss coin until a head: Discrete
2. Time to access webpage: Not discrete
3. Network Routers: Discrete
– Discrete Sample Space - sample space with a finite or countably infinite
number of elements.
Stat 330 (Spring 2015): slide set 1
We try to access a web page and record the time it takes for the webpage
to respond.
2. Record the time for a webpage to respond.
Stat 330 (Spring 2015): slide set 1
The Wall Street Journal tracks the DOW Jones industrial averages.
1. Toss a coin until we get a head.
Elementary Outcome (ω) - an outcome of a random process. Examples:
Record the number of car accidents at an intersection.
Stat 330 (Spring 2015): slide set 1
Components of Random Experiments
Record the result of tossing two coins repeatedly: HH,HT,TH,TH,HH,...
Definition: A random experiment is a process with random outcomes.
Examples of Random Experiments
Stat 330 (Spring 2015): slide set 1
C = Successful transmission
= {DU U, U DU, U U U }
3. Suppose the message is transmitted successfully if at least one router
is up and the recipient’s computer is up.
B = More than 10 seconds
= (10, ∞)
2. Time to access a webpage:
A = first head occurs between 5 and 11 tosses (inclusive)
= {5, 6, 7, 8, 9, 10, 11}
1. Toss coin until a head:
Event (A) - A ⊂ Ω i.e. subset of Ω. (A is a collection of elementary
Components of Probability Experiments (continued)