Every week, Alex receives 1,000 rubles allowance from his parents [ I US dollar 73 Russian rubles ]. He usually spends most of it buying candy. In Alex's favorite candy store, W&W's (a cheap imitation of M&M's) are sold in bulk at 100 rubles per kg, and Aeese's Pieces (knock off Reese's Pieces) are sold at 200 rubles per kg. Alex's Mom is very concerned about this unhealthy habit; she made Alex promise her that he would not buy more than 6 kg ofW&W's (she does not know that he also buys fleese's Pieces ). Let X and Y denote the weight (in kg) ofW&W's and Aeese's Pieces Alex buys, respectively. Let the joint probability density function for ( X, Y) be
f(x,y) 3x+2y no, y20, x<6, IOOx+200y< 1000,
240 zero otherwise. X- W&W's, Y - fleese's Pieces.
Recall (Homework #2):
20 + 8x —x2
fx(x) fy(y) -
192
h) Find the probability that Alex would spend over 540 rubles buying candy.
That is, find 100X+ 200 Y > 540).
i) Find the probability that Alex would spend over 760 rubles buying candy. That is, find 100X+ 200 Y > 760).
j) Find k) Find
l) Suppose we know that Alex bought more than 1.6 kg ofW&W's. What is the probability that he bought more than 3.6 kg of Aeese's Pieces? That is, find 3.6 | x > 1.6 ).
m) Suppose we know that Alex bought exactly 1.6 kg ofW&W's. What is the probability that he bought more than 3.6 kg of Aeese's Pieces? That is, find 3.6 | x = 1.6 ).
n) Find E(Y I X = x), the expected weight of 51eese's Pieces that Alex bought, given that he bought exactly x kg of W&W's.
o) Suppose we know that Alex bought exactly 0.3 kg of fleese's Pieces. What is the probability that he bought more than 3.2 kg ofW&W's? That is, find I Y= O.3 ).
p) Suppose we know that Alex bought exactly 3.0 kg of Aeese's Pieces. What is the probability that he bought more than 3.2 kg ofW&W's? That is, find
q) Find E(X I Y = y), the expected weight ofW&W's that Alex bought, given that he bought exactly y kg of Aeese's Pieces.
FLIP FLIP FLIP
