Question

Roll two fair four-sided dice. Let X and Y be the die scores from the 1st die and the 2nd die, respectively, and define a random variable Z = X − Y (a) Find the pmf of Z. (b) Draw the histogram of the pmf of Z. (c) Find P{Z < 0}. (d) Are the events {Z < 0} and {Z is odd} independent? Why?

Answer 1

a)below is pmf of Z

P(Z=-5) =P(X=1,Y=6) =1/36

P(Z=-4) =P(X=1,Y=5)+P(X=2,Y=6)=2/36

P(Z=-3)=3/36

P(Z=-2)=4/36

P(Z=-1)=5/36

P(Z=0)=6/36

P(Z=1)=5/36

P(Z=2)=4/36

P(Z=3)=3/36

P(Z=4)=2/36

P(Z=5)=1/36

b)

c)

P(Z<0) =P(Z=-5)+P(Z=-4)+P(Z=-3)+P(Z=-2)+P(Z=-1) =5/12

d)

P(Z<0) =5/12

P(Z is odd) =P(Z=-5)+P(Z=-3)+P(Z=-1)+P(Z=1)+P(Z=3)+P(Z=5) =1/2

P(Z<0 and Z is odd) =P(Z=-5)+P(Z=-3)+P(Z=-1) =1/4

since P(Z<0)*P(Z is odd)=(5/12)*(1/2) =5/24 is not equal to P(Z<0 and Z is odd)

therefore event {Z < 0} and {Z is odd} are not independent