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

Bounty: 50

$$IC=mathbb P(sqrt{V} cos(pi U)leq c),$$
$$IS=mathbb P(sqrt{V} sin(pi U)leq c),$$
$$ICS=mathbb P(sqrt{V} cos(pi U)leq c_1, sqrt{V} sin(pi U)leq c_2),$$
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

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 $k>2$.

enter image description here

This may help. Any special case is also useful.

Thanks in advance for any help you are able to provide.

Get this bounty!!!

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