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.

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 2 ac, the continuous random variable Y is uniformly distributed between zero and 3:13. fx) x (dollars) 1. Determine the joint PDF fX'y (as, y). |f0 3w: gs/(93,11): 2. 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)= 3. Find the PDF of Z. Express your answers in terms of z using standard notation. Hint: Start by finding fZlX (z ] x). If 40 80: fz(z) 2 4. What is E[Z]? E[Z] = This gure below describes thejoint PDF of the random variables X and Y. These random variables take values in [0, 2] and [0,1], respectively. At a: = 1, the value of thejoint PDF is 1/2. fx.Y (Ii 11) = % 0.3 0'6 0'4 0'2 1.Are X and Y independent? 0 Yes ONO 2. Find fx (3). Express your answers in terms of m, using the standard notation. IfO 1/2: fY|X(y I 0-5) = 4. Find fX|y(x [ 0.5). Ifl/2 3/2: fX|Y(1' l 0-5) = 5. LetR 2 XY and letA be the event that {X