# #StackBounty: #probability Find \$mathbb P(sqrt{V} cos(pi U)leq c)\$, \$mathbb P(sqrt{V} sin(pi U)leq c)\$

### Bounty: 50

Find
$$IC=mathbb P(sqrt{V} cos(pi U)leq c),$$
$$IS=mathbb P(sqrt{V} sin(pi U)leq c),$$
and
$$ICS=mathbb P(sqrt{V} cos(pi U)leq c_1, sqrt{V} sin(pi U)leq c_2),$$
where $$Vsim chi^2_{k}$$ and $$Usim Beta(a,b)$$, that is,
$$f_V(v)=frac{1}{Gamma(alpha) 2^{alpha/2}} v^{alpha/2-1} e^{-frac{v}{2}}1_{v>0}$$ and
$$f_U(u)=frac{1}{Beta(a,b)}u^{a-1}(1-u)^{b-1}1_{(0,1)}(u)$$.

For a special case $$a=b=1$$, $$alpha=2$$ , the distribution of $$sqrt{V} cos (pi U)$$ is standard normal, the distribution of $$sqrt{V} sin (pi U)$$ is standard normal too. The Box-Mueller transformation is special case of this transformation(for $$k=2$$ and $$a=b=1$$). The random variable $$Z=sqrt{V} cos(pi U)$$, has a good property. By a simple simulation, it can be symmetric for $$a=b$$, and otherwise
is asymmetric. It is also be bimodal for $$k>2$$.

This may help. Any special case is also useful.