Expected coordinates of a uniform point chosen out of a triangle with vertices $(x1,y1), (x2,y2), (x3,y3)$

Question

I think the answer is probably $(1/3*(x1+x2+x3), 1/3*(y1+y2+y3))$.

But is there a simply way to derive it? Thanks for your help.

Answer 1

The expected value of a random variable distributed uniformly over a bounded region of $A \subset {\mathbb R}^2$ (same goes for ${\mathbb R}^n$) is its centroid. That comes directly from its definition. For each coordinate:

$$E[X_i] = \int_A x_i f_{X_1,X_2}(x_1,x_2) dx_1 dx_2=\frac{\int_A x_i f_{X_1,X_2}(x_1,x_2) dx_1 dx_2}{\int_A f_{X_1,X_2}(x_1,x_2) dx_1 dx_2}$$

Now, we know that the centroid of a planar triangle in is given by the average of its vertex coordinates. To deduce that from the integral is doable, though not terribly easy or nice. For example.