Problem $4.$ Sophia's vacation

$5\sqrt[\phantom{2}]{7}$ points $($graded$)$

Sophia is vacationing in Monte Carlo. On any given night, she takes $x$ dollars to the casino and returns

with $Y$ dollars. The random variable $x$ has the PDF shown in the figure. Conditional on $x=x,$ the

continuous random variable $Y$ is uniformly distributed between zero and $3x.$

Determine the joint PDF $fx,Y(x,y).$

$If03x$:

$fX,Y(x,y)=$

On any particular night, Sophia makes a profit $Z=Y-x$ dollars. Find the probability that Sophia

makes a positive profit, that is$,$ find $P(Z>0).$

$P(Z>0)=$

Find the PDF of $Z.$ Express your answers in terms of $z$ using standard notation.

Hint: Start by finding $f_{Z|x}(z|x).$