Random vectors

Random Vectors are simply groups of random variables, arranged as vectors. E.g.:

In general, all of the previous results can be applied to random vectors as well as to random scalars, but vectors allow someinteresting new results too.

Example - arrows on a target

Suppose that arrows are shot at a target and land at random distances from the target centre. The horizontal and verticalcomponents of these distances are formed into a 2-D random error vector. If each component of this error vector is anindependent variable with zero-mean Gaussian pdf of variance $\sigma ^2$ , calculate the pdf's of the radial magnitude and the phase angle of the error vector.

Let the error vector be

The 2-D pdf of $X$ depends only on $r$ and not on $\theta$ , so the angular pdf of the error vector is constant over any $2\pi$ interval and is therefore $$f_{\Theta }(\theta )=\frac{1}{2\pi }$$ so that $$\int_{-\pi }^{\pi } f_{\Theta }(\theta )\,d \theta =1$$