1.
A new startup company has developed a top-of-the line wind turbine for generating electricity. The company provides the following reliability assurances: The time (in years) until first failure of the system, is given by random variable X which is distributed according to
1
fx (x) = u(x − 1)
where u(·) is the standard unit step function. Evidently, there is no failure in the first year, which is of course covered by warranty...
The first failure is usually repairable so the system can be put back into service, and the company provides further assurances. Let random variable Y denote the overall lifetime of the system (where, clearly, Y > X). It has been established that when X takes the value x, the system overall lifetime Y is distributed according to x/y2, for y≥ 2. Find:
a) The joint density fxy(x, y) and the marginal density of the system overall lifetime fr (y).
b) The conditional density of the time to first failure, given the system overall lifetime: fxy (xy).
c) The probability that the overall lifetime will be at least three times the period until first failure, i.e., PY > 3X].
2. A Powerful detector measure the frequency of radiation
virtually the entire electromagnetic spectrum, ranging from 1 kHz (extremely low frequency radio waves used to communicate with submarines) through the visible spectrum, and all the way up to frequencies of over 1020 Hz (Gamma rays). Let random variable Z be the frequency measured in kHz. Its PDF is:
「z(2) = au(2 (-1) -1),
where u(·) is the standard unit step function.
a) The wavelength is given by random variable Y = cZ-1, where constant e denotes the speed of light (in appropriate units). Find its PDF, fy (y).
b) Given the extensive electromagnetic spectrum, it is often useful to plot the frequency measurements in log scale. Let X = log(Z), and find its PDF, fx(x).
c) Are X and Y uncorrelated and/or independent? Prove your answer mathematically, and explain why it should have been expected.
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