Math 21a, Computer Science Section, Problems for Week 11

  • Due in class on Tuesday, December 4
  • For problems 4-6, expect to encounter functions that are described by different formulas on different intervals.
  • For questions 13-14, you might enjoy coding the answer in C.

  1. Apostol, section 14.16, problem 2.
  2. Apostol, section 14.16, problems 8 and 9
  3. Apostol, section 14.16, problem 14
  4. Apostol, section 14.18, problem 1, parts a,c,e and f
  5. Apostol, section 14.22, problem 3
  6. Apostol, section 14.24, problem 1
  7. Apostol, section 14.27, problem 3. Consider just the continuous case. You may assume any well-known general properties of integrals.
  8. Apostol, section 14.27, problems 4 and 5.
  9. Apostol, section 14.27, problem 10.
  10. You have acquired two atoms of a radioisotope that decays in accordance with an exponential distribution whose decay constant is l.
  11. Let X and Y be random variables corresponding to the time after which atoms 1 and 2 respectively decay.
    • a. Find a probability density for the random variable X - Y.
    • b. What is the expectation of | X - Y|?
  12. Apostol, section 14.27, problem 12.
  13. Let X and Y be independent random variables, each with a standard normal distribution. Let Z = X2 + Y2.
    • a) Find a density function fZ(t) for the random variable Z, and shetch a graph of this function.
    • b) Find the expectation of Z.
    • c) Find the value of t for which the density fZ(t) has its maximum value.
  14. Suppose you write a function in C that does the following:
    • 1). Generate a random real number x with a uniform distribution in (0, c]
    • 2). Return the value y = - log(x) (natural logarithms, of course)
    What is a probability density fY(y) for the output of this function?
  15. Assume that you have a function that generates random floating-point numbers in [0,1] with a uniform distribution.
    • a) Invent a method to generate random floating-point numbers Y in [0,4] with a density function proportional to the square root of Y.
    • b) What is the probability that a random number generated by this method will be less than 1? There are two ways to answer this question -- using them both will provide a check of your answer.