Statistics -> Variance and Linear Combinations

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The discussion revolves around calculating variance and expectations in probability problems. The user calculated the variance of their game involving rolling two dice to be approximately 2.06, with an expected value of 1.94, and expressed uncertainty about preferring a larger variance for potentially higher winnings. They also struggled with linear combinations, specifically finding the probability density function (PDF) and expectation for a random variable defined by a cubic transformation. Despite attempts to derive the new PDF and expectation, the user remains confused about the correct approach and calculations. Clarity on these statistical concepts is essential for accurate problem-solving in variance and linear combinations.
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Having a lot of trouble with a particular problem in the topic of variance. The problem is:

"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.
 
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I guess I need to find the expectation value for the second problem first before doing anything, so:

integral of x*2x = integral of 2x^2 = 2x^3/3 between 1 and 0 = 2(1)^3/3 = (2/3) = .75

So from there I would just say (.75)^3 as the answer for the expectation value for Y = X^3?
 
Anyone? Still isn't making sense to me.
 
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
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