How do you calculate distance to origin for a random point in {−3,−1,1,3}×{−4,−2,2,4}

probability

The question is if this sample space is uniform:

{distance to origin} for a random point in {−3, −1, 1, 3} × {−4, −2, 2, 4}

How do you calculate each distance to origin?

Best Answer

Welcome!

A cartesian product $A\times B$ is a set which contains a pair of elements of the set $A$ as first element, and elements of $B$ as second element. Formally, $$A\times B=\{(x,y)\mid \text{$x\in A$ and $y\in B$}\}.$$ So for example, if you have $\{0,1\}\times\{0,4\}$ you will get $\{0,1\}\times\{0,4\}=\{(0,0),(0,4),(1,0),(1,4)\}$.

Once you find all the elements of the cartesian product, you can use the formula of distance between two points in 2D: if $P=(x,y)$ and $Q=(z,t)$ are two points in two dimensions, then the distance between them is $$\operatorname{dist}(P,Q)=\operatorname{dist}(Q,P)=\sqrt{(x-z)^2+(y-t)^2}.$$ Do these calculations for every point... or think about a possible symmetry between e.g. points $(-3,4)$ and $(3,4)$.

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