Question

Paul is vacationing in Monte Carlo. On any given night, he 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 2x.


Determine the joint PDF fX,Y(x,y).

If 0<x<40 and 0<y<2x,

fX,Y(x,y)=
MY ANSWER : 1/1600
If y<0 or y>2x,

fX,Y(x,y)=
MY ANSWER:0
On any particular night, Paul makes a profit of Z=Y−X dollars. Find the probability that Paul makes a positive profit (i.e., P(Z>0)):


MY ANSWER :0.5
Find the PDF of Z. Express your answers in terms of z using standard notation . Hint: Start by finding fZ|X(z|x).

If 0<z<40, fZ(z)=
couldn't answer

If −40<z<0, fZ(z)=
couldn't answer

If z<−40 or z>40, fZ(z)=
couldn't answer
What is E[Z]?

E[Z]=
couldn't answer

Answer 1

what if we had 3x instead of 2x