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I'm now learning mathematical-statistics, and I learned a lot of example, like

" $X $and $Y$ are two independent gamma distribution, please prove that the addition of two gamma distribution is still a gamma"

"What is the p.d.f of $X/Y$?"

I want to know, in practice, when will we need to use a distribution after +-*/ the several different distributions?

I mean, a random variable that follows gamma distribution is the time interval between a specific number of occasional events, such as time interval between n car incidences.

And I have no idea for the purpose of$X+Y $, $X/Y$

Thanks

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    $\begingroup$ This is a bit trivial and maybe not really about statistics. Often your variable of interest is (simply) an arithmetic operation of two (or many) other variables (it is by definition; it is given). See for instance a Galton Board where a bean is falling down encountering many pins and every pin is making the bean move left or right with some (approximately) independent probability (e.g. a fifty-fifty Bernouilli variable and their sum will follow a binomial distribution). $\endgroup$ – Sextus Empiricus Nov 28 '19 at 7:34
  • $\begingroup$ Possibly your question was about the reasoning behind a t-distribution or F-distribution in hypothesis testing? $\endgroup$ – Sextus Empiricus Nov 28 '19 at 7:37
  • $\begingroup$ that is not very clear to me what you mean precisely. Did you think about broswing though examples first? Maybe this helps you to explain your question. There are many examples here that explain when we need to use a product or division. An intuitive example from myself that comes to my mind is here and it relates the joint distribution of X and Y to a ratio distribution for Z=X/Y. (see the image in that answer) $\endgroup$ – Sextus Empiricus Nov 28 '19 at 10:14
  • $\begingroup$ @Sextus Empiricus Oh, thank you! I just had a hard time for understand things in an abstract way. $\endgroup$ – S.F. Yeh Nov 28 '19 at 10:19
  • $\begingroup$ @ Sextus Empiricus And yes, I think the reasoning behind t distribution is pretty relevant to the reason for my confusing. t is a combined distribution of normal and chi-square, not really understand why it is so. I think I better do more work on this. Thank you again! $\endgroup$ – S.F. Yeh Nov 28 '19 at 10:25
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Here's an example. Suppose that I bet 10 dollars on whether a coin toss comes up heads. If it comes up heads, I get \$20, and if I comes up tails, I give up \$10. At the same time, I bet \$10 on whether a pair of dice turns up "snake eyes" (two 1's). If it comes up snake eyes, I get \$20, and if it doesn't, I lose \$10.

What is the expectation of my winnings? If $X$ is the random variable for the the amount of money I win on the coin toss (this amount will be negative if I lose), and $Y$ is the amount of money I win on the dice toss, the expectation of my winnings is $\mathsf{E}(X + Y) = \mathsf{E}X + \mathsf{E}Y$. This could also be expressed in terms of subtraction, by defining $X$ and $Y$ in a different way.

Obviously, there are more ways to use this idea than by calculating expectations. The idea also applies to many things other than simple games of chance, of course. For example, suppose that a bird eats a berry, and it will either provide the bird with a certain amount of energy, or it will reduce the bird's energy, if the berry is slightly poisonous. After that, the bird tries to catch an insect. It will gain energy if it catches the insect quickly, but if it is difficult to catch the insect, it might lose more energy. Given certain probabilities for the outcomes of eating the berry and the outcomes of pursuing the insect, what the expectation of the energy increase?

Suppose that there is a need to sum many identical random variables (perhaps many instances of the bird eating berries), but that the number of random variables depends on another variable. Maybe the number of berries eaten depends on when a predator arrives. This could be represented by multiplying the random variable for berry-eating, $X$, by the random variable for the arrival time of a predator, $Z$.

There are cases where division matters. Suppose that if the bird is ill, then nutritional value of the berries is reduced by dividing the energy by a random quantity $W$ that reflects how ill the bird is.

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