Steve is a self$-$employed IT consultant. From past experience, his monthly income $Y($in US$$1000)$ is found to follow the Chi$-$square distribution with degrees of freedom $6.$ Given $Y=y,$ his expenses $x($in US $$1000)$ in the corresponding month has the following conditional probability density function $(pdf)$ :

$xZ=Y-xZ=\frac{1}{2}$Cov$(x,Y)$$$43,000$