Consider the joint pdf

f$($x$,$ y$)=3($x$2+$ y$)/11$ for $0<=$ x $<=2,0<=$ y $<=1.$

$($a$)$ Plot in python the joint pdf f$($x$,$ y$).$

$($b$)$ Show that the marginal pdf of X is

fx$($x$)=3(2$x$2+1)/22$ for $0<=$ x $<=2.$

Plot in pyhton fx$($x$).$

$($c$)$ Derive the marginal pdf of Y $,$ fy$($y$).$ Plot in pyhton fy$($y$).$

$($d$)$ Derive the conditional pdf of Y given X$,$ f$($y$|$x$).$ Plot in python f$($y$|$x$)$ for X $=0,1,2.$

$($e$)$ Derive the CEF of Y given X$.$

$($f$)$ Calculate E$($X$),$ E$($Y $),$ E$($X$2),$ E$($Y $2),$ E$($XY $),$Var$($X$),$Var$($Y $),$ Cov$($X$,$ Y $).$

$($g$)$ Find the best linear predictor of Y given X$.$

$($h$)$ Plot in Pythom the CEF and BLP as a function of X$.$