# Sample spaces and events

01/27/2021

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## Outline

• The following topics will be covered in this lecture:

• Random experiments
• Discrete and Continuous Sample Spaces
• Events and set notation
• Permutations and combinatinos

## Random experiments

• Suppose we conducting an experiment: measure the current in a thin copper wire.
• The measurements can differ slightly because of small variations in variables that are not controlled in our experiment, e.g.:
• changes in ambient temperatures;
• slight variations in the gauge;
• impurities in the chemical composition of the wire;
• current source drifts; and/or
• measurement errors.
• No matter how carefully an experiment is designed and conducted, the variation is almost always present Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition

Our goal is to understand, quantify, and model the type of variations that we often encounter.

• Ours models incorporate uncontrollable inputs (noise) that combine with the controllable inputs to produce the output of our system.
• Because of the uncontrollable inputs, the same settings for the controllable inputs do not result in identical outputs every time the system is measured.

### Random experiments

Random experiment: an experiment that can result in different outcomes, even though it is repeated in the same manner every time.
• Example: in the design of a communication system, such as a voice communication network, the information capacity available to serve individuals using the network is an important design consideration.
• For voice communication, sufficient external lines need to be available to meet the requirements of a business.
• Assuming each line can carry only a single conversation, how many lines should be purchased?
• If too few lines are purchased, calls can be delayed or lost.
• The purchase of too many lines increases costs.
• Increasingly, design and product development is required to meet customer requirements at a competitive cost.
• In the design of the voice communication system, a model is needed for the number of calls and the duration of calls.
• Even knowing that, on average, calls occur every five minutes and that they last five minutes is not sufficient.
• If calls arrived precisely at five-minute intervals and lasted for precisely five minutes, one phone line would be sufficient.
• However, the slightest variation in call number or duration would result in some calls being blocked by others.
• A system designed without considering variation will be woefully inadequate for practical use.
• Our model for the number and duration of calls needs to include variation as an integral component.

### Sample spaces

The set of all possible outcomes of a random experiment is called the sample space of the experiment. The sample space is denoted as $$S$$.
• Consider an experiment that selects a cell phone camera and records the recycle time of a flash (the time taken to ready the camera for another flash).

• Because the time is positive, it is convenient to define the sample space as simply the positive real line $S=R^+ = \{x|\: x>0 \}$

• If it is known that all recycle times are between 1.5 and 5 seconds, the sample space can be $S = \{x|\: 1.5 < x < 5 \}$

• If the objective of the analysis is to consider only whether the recycle time is low, medium, or high $S=\{low,\:medium,\:high \}$

• If the objective is only to evaluate whether or not a particular camera conforms to a minimum recycle-time specification $S=\{yes,\:no \}$

• The sample space $$S$$ depends on the kinds of measurements we are taking as above.

### Discrete and Continuous Sample Spaces

• A sample space is continuous if it contains an interval (either finite or infinite) of real numbers.

• Recall the camera flash example, $$S = R^+$$ is an example of a continuous sample space.
• The unit of measurement can be arbitrarily sub-divided (seconds, miliseconds, etc.) and the measurement still makes sense.
• A sample space is discrete if it consists of a finite or countable infinite set of outcomes.

• Recall the camera flash example, $$S = \{yes,\: no\}$$ is a discrete sample space.
• The unit of measurement above cannot be arbitrarily sub-divided making the sample space discrete.
• The best choice of a sample space depends on the objectives of the study.

### Discrete and Continuous Sample Spaces

• Suppose that the recycle times of two cameras are recorded.

• The extension of the positive real line $$R$$ is to take the sample space to be the positive quadrant of the plane $S=R^+ \times R^+$

• Measurements would come in pairs of time units, one piece of data for each camera.

• If the objective of the analysis is to consider only whether or not the cameras conform to the manufacturing specifications, either camera may:

• (yes=y) conform; or
• may not conform (no=n)
• Then the sample space can be represented by the four outcomes: $S=\{yy,yn,ny,nn\}$

• If we are interested only in the number of conforming cameras in the sample, then the sample space can be $S=\{0,1,2\}$

### Tree diagrams

• Consider an experiment in which cameras are tested until the flash recycle time fails to meet the specifications. The sample space can be represented as $S=\{n, yn, yyn, yyyn, yyyyn, \text{and so forth}\}$
• Sample spaces can also be described graphically with tree diagrams when a sample space can be constructed in several steps or stages.
• Each of the $$n_1$$ ways of completing the first step as a branch of a tree.
• Each of the ways of completing the second step can be represented as $$n_2$$ branches starting from the ends of the original branches, and so forth.
• EXAMPLE: messages in a communication system are classified as to whether they are received within the time specified by the system design. The following a tree diagram represents the sample space of possible outcomes for three messages. Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition

• Each message can be received either on time or late
• Stage 1: message 1
• $$n_1=2$$ ways of completing message
• Stage 2: message 2
• $$n_2=2$$ ways of completing message
• Stage 3: message 3
• $$n_3=2$$ ways of completing message

### Events

• An event is a subset of the sample space of a random experiment.
• Events are subsets, we can use basic set operations
• We can use Venn diagrams to represent a sample space and events in a sample space.
Union of two events is the event that consists of all outcomes that are contained in either of the two events.
Denoted $$A \cup B$$ Intersection of two events is the event that consists of all outcomes that are contained in both of the two events.
Denoted $$A \cap B$$ Complement of an event in a sample space is the set of outcomes in the sample space that are not in the event.
Denoted $$A'$$ or $$A^c$$ ### Camera flash events

• Recall the camera flash example with sample space $$S=\{yy,yn,ny,nn\}$$.
• Suppose that the subset of outcomes for which at least one camera conforms is denoted as $$E_1$$ $E_1 = \{yy, yn, ny\}$
• When the cameras do not conform, denote as event $$E_2$$, contains only the single outcome, $E_2 = \{nn\}$
• Other events are
• $$E_3 = \emptyset$$, the null set
• $$E_4 = S$$, the sample space
• Question: If $$E_5 = \{yn, ny, nn\}$$, what is …
• $$E_1 \cup E_5 =$$?
• $$E_1 \cup E_5 =S$$
• $$E_1 \cap E_5 =$$?
• $$E_1 \cap E_5 =\{yn,ny\}$$
• $$E_1' =$$?
• $$E_1' =\{nn\}$$