3. You start a job as a quant. On any given day, you randomly make 5 X $1k for the fund, with f3 (5) = 0.015e0'15 for s > 0 and zero otherwise (i.e., you never lose money). (a) (3 pts) Draw a rough sketch of f5. i. At what 5 does f5(s) peak? This is called the mode of the PDF. ii. Show that the mean (another word for E [5]) is n_ot equal to the mode. iii. What continuous RV. you know has its mean always equal to its mode? (b) (1 pt) If you make 5 = s x $1k for the fund, you receive a uniformly random bonus U between 0% and 10% of s. Interpret this as a conditional PDF fu| 5(u Is) and express it mathematically. Then combine it with fs(s) to express the joint PDF fs!u(S, u). (c) (1 pt) On the (5,1,1) real plane, outline where fs,u is non-zero. On the same plot, shade the event U > 1. (d) (2 pts) Use (b) and (c) to help calculate the probability that you make more than $1k. [Hint: To get this probability. you need to integrate the joint PDF. (b) gives you what to integrate. (c) helps you find the range of integration.]