Question

X, Y and Z are Bernoulli random variables with the following joint distribution:

P(x, y, z) =

.2 if (x, y, z) = (0, 0, 0)

.1 if (x, y, z) = (0, 0, 1)

0 if (x, y, z) = (0, 1, 0)

.1 if (x, y, z) = (0, 1, 1)

.1 if (x, y, z) = (1, 0, 0)

0 if (x, y, z) = (1, 0, 1)

.2 if (x, y, z) = (1, 1, 0)

.3 if (x, y, z) = (1, 1, 1)

a) Find the marginal distribution of Z.

b)Find the joint distribution of X and Y , conditional on Z = 1.

c) Find the correlation coefficient between X and Y , conditional on Z = 1.

Answer 1

a)

add probabilities for Z=0

and Z=1

P(Z=0) = 0.2+0 + 0.1 + 0.2 = 0.5

P(Z=1) = 0.1 + 0.1 + 0 + 0.3=0.5

z	0	1	total
P(Z=z)	0.5	0.5	1

b)

row represent X, column represent Y

condition Z=1

P(Z=1)=0.5

X/Y	0	1	total
0	0.1/0.5=0.2	0.1/0.5=0.2	0.4
1	0/0.5=0	=0.3/0.5=0.6	0.6
total	0.2	0.8	1

c)

correlation coefficient between X and Y , conditional on Z = 1

X	Y	P(X,Y)	X*Y*P(X,Y)	X*P(X)	(X-E[X])²P(X)	Y*P(X,Y)	(Y-E[Y])²P(Y)
0	0	0.2	0	0	0.072	0	0.128
0	1	0.2	0	0	0.072	0.2	0.008
1	0	0	0	0	0	0	0
1	1	0.6	0.6	0.6	0.096	0.6	0.024
total			0.6	0.6	0.24	0.8	0.16

co variance=E[XY]-E[X]E[Y]=ΣXYP(X,Y) + ΣXP(X)+ΣYP(Y) = 0.6 - 0.6*0.8 = 0.12

std dev of X=√(X-E[X])²P(X)=√0.24

std dev of Y=√(Y-E[Y])²P(Y) =√0.16

correlation=cov(x,y)/(std of x*std dev of y) = 0.12/(√0.24*√0.16) = 0.6124