Let X be the price of a meal at a local diner X810 and 12 an

Let X be the price of a meal at a local diner (X=$8,$10, and $12) and let Y be the amount of tip left per meal (Y=$1.50, $2, or $2.50). The joint pdf of X and Y is given by

a.) Find f(X=10,Y=2) and interpret what this value represents.

c.) Find the marginal distribution of X and Y.

d.) Determine if X and Y are independent RVs or not. Explain why or why not.

Solution

a) f(X=10,Y=2)=0.135 . it is the probability that the price of a meal at a local diner is 10$ AND amount of tip left after per meal is 2$.

b)P[X<=10,Y<=2]=P[X=8,Y=1.5]+P[X=8,Y=2]+P[X=10,Y=1.5]+P[X=10,Y=2]=0.3+0.12+0.15+0.135=0.705 [answer]

c)marginal of X

P[X=8]=0.3+0.12+0=0.42   P[X=10]=0.15+0.135+0.025=0.31   P[X=12]=0.03+0.15+0.09=0.27

hence marginal of X is

X:          8                10                 12

P[X=x]:     0.42             0.31              0.27

P[Y=1.50]=0.3+0.15+0.03=0.48   P[Y=2]=0.12+0.135+0.15=0.405   P[Y=2.5]=0+0.025+0.09=0.115

so marginal of Y

Y:                       1.5                  2                2.5

P[Y=y]:               0.48                0.405          0.115

d) P[X=8]*P[Y=1.5]=0.42*0.48=0.2016 which is not equal to 0.3 which is P[X=8,Y=1.5]

hence X and Y are not independent.

e) P[X<=10 | Y=2]=P[X<=10,Y=2]/P[Y=2]=(P[X=8,Y=2]+P[X=10,Y=2])/P[Y=2]=(0.12+0.135)/0.405=0.6296 [answer]