Participants from a day-long conference choose from 2 lunch specials: Sirloin tips $10 or Spinach lasagna $5 and from 3 dinner specials: Lobster $25, Crab legs $20, or Thai Stir-fried vegetables $10. For a randomly selected participant, the joint probability distribution p(x; y) for X = the cost of lunch and Y = the cost of dinner is given by p(x; y) : X/Y   $10 $20 $25  $5    0:30 0:20 0:10  $10  0:20 0:10 0:10 a) Find the marginal probability density functions pX(x) and pY (y) for X and for Y . b) Find the expected cost of meals for a randomly selected participant E[X + Y ]. c) Compute the correlation coe¢ cient between X and Y .

Participants from a day-long conference choose from 2 lunch specials: Sirloin tips $10 or Spinach lasagna $5 and from 3 dinner specials: Lobster $25, Crab legs $20, or Thai Stir-fried vegetables $10. For a randomly selected participant, the joint probability distribution p(x; y) for X = the cost of lunch and Y = the cost of dinner is given by p(x; y) :
X/Y   $10 $20 $25
 $5    0:30 0:20 0:10
 $10  0:20 0:10 0:10

a) Find the marginal probability density functions pX(x) and pY (y) for X
and for Y .

b) Find the expected cost of meals for a randomly selected participant

E[X + Y ].

c) Compute the correlation coe¢ cient between X and Y .

d) Are X and Y independent? Explain.

e) Find the mean and variance of the random variable W = 3X - Y .