Consider a covariant stationary stochastic process \{X (t)\} for which the joint probability density function of X (t) and the derivative \dot{X}(t) at the same instant of time is given by
p_{X(t)\dot{X}(t)} (u,v)=\frac{1}{\pi 2^{1/2}u} \exp \left(-u^{2}-\frac{v^{2}}{2u^{2}} \right)Find the marginal probability density function p_{X(t)}(u) and the conditional probability density function for the derivative, p_{\dot{X}(t)}[v\mid X(t)=u] . Determine whether X(t) is a Gaussian random variable and whether \{X(t)\} is a Gaussian stochastic process.