Every week, Alex receives 1,000 rubles allowance from his parents [ 1 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 51eese'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
3x+2y x 2 0, y20, x<6, IOOx+200y< 1000,
240 zero otherwise. X- W&W's, Y-Aeese's Pieces.
Recall (Homework #3 ):
fylx(ylx)
30 (2 +X)
Find var(Y I x=x). You do not have to simplify the final answer.
Find the probability distribution of the total weight of candy W = X + Y.
Let T = 100 X + 200 Y denote the amount Alex spends on candy.
Find the cumulative distribution function of T, t ).
Find E(T), the average (expected) amount Alex spends on candy.
x
Let V- Find the cumulative distribution function of V, Fv(v).
You are welcome to use a computer and/or calculator on any problem to evaluate any integral. For the supporting work, you should include the full integral ( with the function and the bounds) and the answer. For example,
3 4
J u 2 du — x3 x 2 y dy dx = 32,dx dy 2
0
