Compute the expected value of a brownian motion

Question

Suppose $X(t)$ is a brownian motion. Compute $E[X(1)X(5)X(7)]$.

I know that the brownian motion has independent increments, so if we could write $X(1)X(5)X(7)$ as such, then we could use the properties of the expected value.

Any suggestions? Is there another way?

Answer 1

We know the distribution of the vector $V:=(X(7)-X(5), X(5)-X(1), X(1))$, since the increments are independent. Defining the function $f\colon (x,y,z)\mapsto (x+y+z)(y+z)z$, the wanted expectation is $\mathbb E\left[f(V)\right]$.