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"Suppose you are organizing a game where you charge players $2 to roll two dice and then you pay them the difference in scores. What is the variance in your profit from each game? If you are playing a game in which you have positive expected winnings, would you prefer a small or large variance in the winnings?"

I already calculated the variance (more details of the problem were not mentioned b/c I already calculated the variance) and it was around 2.06 and the expected value (mean) is 1.94. I would guess a larger variance because then you can take a chance at trying to get higher values while knowing that you have a good chance of getting a positive income in the end. But I keep second-guesing myself of that and I am just not sure, lol.

Also, having a lot of trouble with linear combinations. For example, I have a problem like this:

Suppose that the random variable X has a probability density function of f(x) = 2x for 0 <= x <= 1. Find the PDF and the expectation of the random variable Y in the following cases:

a. Y = X^3

There are 3 more parts but I can do those myself if I can just figure out how to do one. I just have no idea where to start. I read the book and the notes and still having trouble figuring out what to do here. I guess I need to say for example:

f(x^3) = (2x)^3 = 8x^2 would be the new PDF?

and ...

E(x^3) = (E(x))^3 = ... ?

I don't even know what I am doing ... lol.