Question

Let X take the value 0 if a child under age 5 uses no seat belt, 1 if the child uses an adult seat belt, and 2 if the child uses a child’s car seat, and it takes these values with probabilities .42, .11 and .47, respectively. Let Y take the value 1 if a child survived a motor vehicle accident and 0 otherwise. A study undertaken by the National Highway Traffic Safety Administration resulted in the following conditional distribution of Y given X: 1 0 P(Y = y|X = 0) .61 .39 P(Y = y|X = 1) .72 .28 P(Y = y|X = 2) .92 .08 (a) Find the marginal pmf of Y, (b) tabulate the joint distribution of X and Y, and (c) determine if X and Y are independent and justify your answer.

Answer 1

a)

marginal pmf of Y at Y=0

P(Y=1) = P(Y=1|X=0)*P(X=0)+P(Y=1|X=1)*P(X=1)+P(Y=1|X=2)*P(X=2)

=0.61*0.42+0.72*0.11+0.92*0.47 = 0.7678

P(Y=2)=P(Y=2|X=0)*P(X=0)+P(Y=2|X=1)*P(X=1)+P(Y=2|X=2)*P(X=2)

=0.39*0.42+0.28*0.11+0.08*0.47=0.2322

Y	1	0	total
P(Y)	0.7678	0.2322	1

b)

X/Y	1	0	total
0	0.2562	0.1638	0.42
1	0.0792	0.0308	0.11
2	0.4324	0.0376	0.47
Total	0.7678	0.2322	1

c)

for independent events P(X=x,Y=Y)=P(X=x)*P(Y=y)

P(X=0)=0.42

P(Y=0)=0.2322

P(X=0,Y=0)=0.1638

P(X=0)*P(Y=0) = 0.0975

since,P(X=0)*P(Y=0)╪P(X=0,Y=0)

so, X and Y are not indepednent