Theoretical distribution of Y(x) and Elipson

I'm a bit confused with this question. So the equation is Y=aX+Ei where:

a: can be positive integer

X: 1-20.

Ei: Uniform distribution of integers from [-9, 9].

It goes on to ask, show and discuss the theoretical distribution of Y|X and Ei.

Well, wouldn't the theoretical distribution of Ei be uniformed? I'm not sure how am I suppose to show it? I'm not sure what the theoretical distribution of Y|X would be. From what the professor said, it should be uniformed too, but that doesn't really make any sense. It would make sense if x was uniformly distributed too, but x is just integers from 1-20.

Thanks for your help!

Re: Theoretical distribution of Y(x) and Elipson

Hey Linnus.

What is the distribution of X? Is it just uniform as well or some other distribution?

Re: Theoretical distribution of Y(x) and Elipson

Hi Chiro,

X is just 1,2,3,4,5,...all the way to 20 (is there a name for this type of distribution?). So the relationship between Y is X is how the the quality of the product goes up with increasing cost (x being the cost). Thanks!

Re: Theoretical distribution of Y(x) and Elipson

Is it a random variable or is it a deterministic variable that can take on those values? Also if it is random and they all have the same choice, the distribution is a discrete uniform (as opposed to a continuous uniform distribution).

Re: Theoretical distribution of Y(x) and Elipson

X is a deterministic variable. Which means X must be 1,2,3,4,5,6...so on all the way to 20. Its given to you and set. You can't change it. Its not determined by picking a number out of a certain distribution. This is why I'm confused - how would Y|X have a uniform distribution (or any distribution at all) if the the X variable is not determined from a certain distribution.

But the professor often gets confused in the class due to his age...so he could be mistaken.

Thanks for your help.

Re: Theoretical distribution of Y(x) and Elipson

So your random variables have the relation Y = aX + Ei.

In probability we have P(Y|X) = P(X|Y)*P(Y)/P(X) (You can look up Bayesian probability for this but its derived by using P(X|Y) = P(X and Y)/P(Y) and P(Y|X) = P(X and Y)/P(X)).

Now consider your random variable Y: if E_i is a discrete uniform in [-a,a] then Ei + C where is a discrete uniform in [-a+C,a+c] where all values are integers. So Y is a discrete uniform distribution.

X is a deterministic variable so what-ever value this takes on it will always have a probability of 1 (deterministic is just a special case of non-deterministic) so P(X) = 1.

So we know P(Y) and P(X) and now P(X|Y) is always 1 since X is deterministic and does not change at all since X does not change given a Y (since it is constant).

So now we have finally P(Y|X)

= P(X|Y)*P(Y)/P(X)

= P(X)*P(Y)/P(X) [Since P(X|Y) = P(X) as X doesn't ever depend on a particular Y value]

= P(Y)

where Y has a discrete uniform distribution.

Re: Theoretical distribution of Y(x) and Elipson

Thanks for the help. I got it. This class only covers regression so we didn't cover probability but I should have thought of it.

Re: Theoretical distribution of Y(x) and Elipson

Wait, quick question, Y isn't really described by Ei+C where C is a constant. Y is more accurately described by Ei+X where X is a variable that changes. I'm not sure why Y would be considered a constant if it changes with X? Thanks!