- Apostol, section 14.16, problem 2.
- Apostol, section 14.16, problems 8 and 9
- Apostol, section 14.16, problem 14
- Apostol, section 14.18, problem 1, parts a,c,e and f
- Apostol, section 14.22, problem 3
- Apostol, section 14.24, problem 1
- Apostol, section 14.27, problem 3. Consider just the continuous case.
You may assume any well-known general properties of integrals.
- Apostol, section 14.27, problems 4 and 5.
- Apostol, section 14.27, problem 10.
- You have acquired two atoms of a radioisotope that decays in accordance
with an exponential distribution whose decay constant is l.
- 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|?
- Apostol, section 14.27, problem 12.
- 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.
- Suppose you write a function in C that does the following:
What is a probability density fY(y) for the output of this function?
- 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)
- 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.