ChapterZero

I’ve been working my butt off for the past several days trying to complete the latest stochastic processes homework, but that looks like a losing fight, so I’ve pretty much relaxed. Now I’m just doing it because every so often I learn a way to do something I thought was incredibly hard.

Example #1: Let be a continuous random variable, and be a random variable uniformly distributed on ; assume and are independent of each other. Calculate where is a fixed number. That is, find the cdf of the random variable . I’m surprised it’s even possible to get a nice closed form solution to this, and sad to say the particular trick used to get it isn’t from me. It’s from one of the guys in the class, either Nathaniel or Yuan, or both. The basic idea is to:

• Find the pdf of the random variable
• Use the independence of and to find the joint pdf of
• The clever idea is here: use a transformation of random variables argument to find the joint pdf of
• Integrate out to get the marginal distribution for

The answer, after working out all this stuff, is Amazing. What’s even more amazing is that this is only one incidental calculation on the way to solving a larger problem. Perhaps there was a way to avoid this unpleasantness altogether?

Example #2: If are independent uniformly distributed random variables on [0,1], then the pdf of the random variable is This one is easy to show, but nonetheless a neat result. In particular, it means the expected value of the random variable is – not terribly intuitive.

Possibly relevant posts:

• Solution to probability of matrix invertibility (5/18/2006)
• Probablistic Integrals (10/27/2006)
• Integrability of matrix functionals (10/6/2008)