Engineering Probability Class 18 Mon 2018-03-26

Source

1   Exam 2, Thurs 3/29

1.1   Summary

  1. Closed book but a calculator and two 2-sided letter-paper-size note sheets are allowed.
  2. Material is mostly from chapter 4, with maybe some from chapters 1-3.
  3. Questions will be based on book, class, and homework, examples and exercises.
  4. The hard part for you may be deciding what formula to use.
  5. Any calculations will (IMHO) be easy.
  6. Speed should not be a problem; most people should finish in 1/2 the time.

1.2   Material on exam

  1. distributions:

    1. uniform: discrete, continuous

    2. exponential: This is the interarrival time between i.i.d (indep and identically distributed) events, e.g., radioactive decays, or web server hits.

    3. normal

    4. Poisson: This is the probability distribution for the number of events in a fixed time, when each possible event is independent and identically distributed.

      Examples:

      1. number of atoms decaying in a block of radium.
      2. number of hits on your web server.
      3. number of students visiting bursar.

    5. binomial

    6. Bernoulli

    7. Geometric

    For each distribution: pdf/pmf, cdf, mean, variance. (Pages 116, 164).

  2. I might give you a new pdf and ask you to compute the cdf.

  3. conditional probabilities.

  4. Markov and Chebyshev inequalities.

  5. function of a r.v.

  6. Reliability. R(t) = 1-F(t).

  7. MTTF (new) p 190.

    MTTF = $\int_0^\infty R(t) dt$

    Ex. MTTF of U[0,1].

  8. Failure rate.

    r(t)dt is the probability of failing in the next dt.

    $$r(t) = \frac{-R'(t)}{R(t)}$$

    Ex: do on U[0,1].

  9. Reliability.

  10. pdf/cdf of the max/min/sum of 2 r.v.

1.3   Material not on exam

  1. characteristic functions.
  2. moment generating functions.
  3. Matlab.
  4. entropy.
  5. generating random variables.
  6. Chi-square
  7. Weibull.

3   Normal Q function

Table from UD Davis E&CE411, Spring 2009.

4   Iclicker questions

  1. Consider a fair tetrahedral die, with faces labeled 1, 2, 3, 4. What is f(2)?
    1. 1/6.
    2. 1/4.
    3. 1/2.
  2. For the same die, what is F(2)?
    1. 1/6.
    2. 1/4.
    3. 1/2.
  3. Consider the continuous distribution with f(x) = $x^2$ for $0\le x \le \sqrt[3]{3}$. What is F(x)?
    1. $x$
    2. $x^2$
    3. $x^3$
    4. $x^2/2$
    5. $x^3/3$
  4. For that distribution, what is F(1)?
    1. 0
    2. 1/3
    3. 1/2
    4. 1
    5. None of the above.
  5. For the uniform U[0,2] distribution, what's the reliability R(1/2)?
    1. 1/2
    2. 1/4
    3. 3/4
    4. 1
    5. None of the above.
  6. For that distribution, what's the failure rate at 1/2?
    1. 1/2
    2. 1/4
    3. 3/4
    4. 1
    5. None of the above.