M.C. Hammer is selling off part of his very large wardrobe of puffy pants. The market price of each pair of pants depends on both the number of sequins on the pants and their age. Let this price be, in dollars, 20+.01S+2A, where S is the number of sequins, A is the age of the pants. The following table presents elements of the joint distribution of S and A in Hammer's wardrobe:
S=200 S=300 S=400 P(A=a)
A=10 .2 .15 .05 .4
A=15 .1 .3 .2 .6
P(S=s) .3 .45 .25 1
a) Fill in the missing values of the table.
b) Make a new table, where the central cells (these are the cells equivalent to the highlighted cells above) contain the values of the conditional PDFs of S. (That is, there will be two PDFs, one for P(S|A=10) and P(S|A=15).)
S=200 S=300 S=400
S|A= 10 .5 .375 .125
S|A= 15 .166666667 .5 .333333
c) Using b, calculate the expected values of S, conditonal on A. Compare these two expectations. Does the relative magnitudes of the expected values yield any intuition about how the two variables are related?
d) Calculate the E[A], E[S], Var[A], and Var[S].
e) Next calculate the covariance of A and S. Finally, calculate the correlation coefficient.
f) If Hammer selects a pair of pants at random, what is the expected price? What is the variance of the price?