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1. Suppose that 90% of web merchants that sell light bulbs
are sleazy: they'll sell you a light bulb that has a 1/20
chance of burning out on any day. The remainder 10% of web
merchants use high-quality merchandise: their bulbs have a
1/1000 chance of burning out on any given day. Sadly, you
can't distinguish the two types of merchants apart from their
web site -- they all look identical.
I pick a web merchant at random, buy a dozen bulbs from
them, and install the first bulb in my kitchen. Let the
random variable X denote the number of days until it burns out.
When it burns out, I curse loudly, then replace it with another
bulb from the same merchant (because I've got to use them up).
Let the random variable Y denote the number of days until the
second bulb burns out (counting from the day when we replaced
the first bulb).
Question: Are the random variables X and Y independent?
Question: Are we guaranteed that E[Z] = E[X] + E[Y]?