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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]?