# What is the purpose of doing Arithmetic(+-*/) on two different distribution?

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

• 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). – Sextus Empiricus Nov 28 '19 at 7:34
• Possibly your question was about the reasoning behind a t-distribution or F-distribution in hypothesis testing? – Sextus Empiricus Nov 28 '19 at 7:37
• 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) – Sextus Empiricus Nov 28 '19 at 10:14
• @Sextus Empiricus Oh, thank you! I just had a hard time for understand things in an abstract way. – S.F. Yeh Nov 28 '19 at 10:19
• @ 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! – S.F. Yeh Nov 28 '19 at 10:25

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.
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.