Let $A=(M,N)^T$ be a bivariate random variable with joint density defined by
$$f_{M,N}(m,n) = \frac{3}{2 \pi} \sqrt{m^2+n^2}$$ if $m^2+n^2<1$ and $0$ otherwise.
Let $B = (F,G)^T$ be given by
$$\begin{pmatrix} F \\ G \end{pmatrix} = \begin{pmatrix} 2.8 & 1 \\ 0&2 \end{pmatrix} \begin{pmatrix} M \\ N \end{pmatrix}$$ .
How do I find the probability $P(0 \le G \le F)$? I know that I can rewrite this into $P(0 \le 2N \le 2.8M+N)$, but how can I find the limits for this double integration (I suppose double integration is possible here)? Thanks in advance!
You're almost there. $$\begin{eqnarray*}\mathbb{P}(0\leq G \leq F)&=&\mathbb{P}(0\leq2N\leq2.8M+N) \\ &=& \mathbb{P}\big(N\leq 2.8M,N\geq 0\big) \\ &=& \int_0^{\arctan(2.8)} \int_0^1 rf_{M,N}(r\cos \theta,r \sin \theta)\mathrm{d}r\mathrm{d}\theta \\ &=&\frac{3}{2\pi}\int_0^{\arctan(2.8)}\int_0^1r^2\mathrm{d}r\mathrm{d}\theta \\&=& \frac{\arctan(2.8)}{2\pi}\end{eqnarray*}$$