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
Noise variables affect the transformation of inputs to outputs

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.

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 E1 E1={yy,yn,ny}
  • When the cameras do not conform, denote as event E2, contains only the single outcome, E2={nn}
  • Other events are
    • E3=, the null set
    • E4=S, the sample space
    • Question: If E5={yn,ny,nn}, what is …
      • E1E5=?
        • E1E5=S
      • E1E5=?
        • E1E5={yn,ny}
      • E1=?
        • E1={nn}

Hospital Emergency Events

Chart of hospital emergency visits

Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition

  • EXAMPLE: the table on the left summarizes visits to emergency departments at four hospitals in Arizona;
    • people may leave without being seen by a physician, and those visits are denoted as LWBS.
  • The remaining visits are serviced at the emergency department, and the visitor may or may not be admitted for a stay in the hospital.
  • Let A denote the event that a visit is to hospital 1, and let B denote the event that the result of the visit is LWBS.
  • Let’s calculate the number of outcomes in AB, A, and AB
  • Q: what is AB?
    • The event AB consists of the 195 visits to hospital 1 that result in LWBS.
  • Q: what is A?
    • The event A consists of the visits to hospitals 2, 3, and 4 and contains 6991 + 5640 + 4329 = 16,960 visits.
  • Q: what is AB?
    • The event AB consists of the visits to hospital 1 or the visits that result in LWBS, or both, and contains 5292 + 270 + 246 + 242 = 6050 visits.
  • Notice that the last result can also be calculated as the number of visits in A plus the number of visits in B minus the number of visits AB (that would otherwise be counted twice) = 5292 + 953 − 195 = 6050.

Mutually Exclusive Events

Venn diagram of mutually exclusive events

Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition

  • Two events, denoted as E1 and E2 , such that E1E2= are said to be mutually exclusive.
  • This is visualized in the Venn diagram when there is no overlap between the events.
  • Notice, if E1E2=, the issue of double-counting the intersection in E1E2 no-long applies.
    • E.g., if E1 represents all patients admitted to hospital 1 and E2 represents all patients admitted to hospital 2, E1E2 can be counted by all the patients admitted to hospitals 1 and 2.
  • Other properties of sets apply
    • Complement properties (E)=E
    • Commutative laws AB=BAandAB=BA
    • The distributive laws (AB)C=(AC)(BC)and(AB)C=(AC)(BC)
    • DeMorgan’s laws (AB)=ABand(AB)=AB