3.54 Determine wheter the two random variables of Exercise 3.19 are dependent or

Question

3.54 Determine wheter the two random variables of Exercise 3.19 are dependent or independent.

3.19 Let x denote the number of times acertain numerical control machine will malfunction: 1,2 or 3 timeson any gives day . Let Y denote the number of times a technician iscalled on an emergency call. Their joint probability distributionis given as

x/y             1                     2                     3

1               0.05                0.05                0.1

2               0.05                0.2                  0.35

3               0                     0.2                  0.1

a) Evaluate the marginaldistribution of X

P(X=1) = 0.05+0.05+0.1 = 0.20 (we sum the proabilities of thefirst row)

P(X=2) = 0.05+0.20+0.35 = 0.60

P(X=3) = 0+0.20+0.10 = 0.3

b) Evaluate the marginal distributeof Y.

P(Y=1) = 0.05+0.05+0 = 0.10 (sum of the prob of the firstcolumn)

P(Y=2) = 0.05+0.20+0.20 = 0.45

P(Y=3) = 0.10+0.35+0.10 = 0.55

c) Find P(Y=2|X=2).

I just need the answer for question 3.54 I already did question 3.49 and those are my answers for that one.

Explanation / Answer

X and Y are independent if P(x|y) = P(x), for all values of X and Y.

P(X=1) = 0.05+0.05+0.1 = 0.20

P(X=2) = 0.05+0.20+0.35 = 0.60

P(X=3) = 0+0.20+0.10 = 0.3

P(X=1|Y=1)= P(X=1, Y=1) /P(Y=1)= 0.05/0.10=0.5

P(X=1|Y=2)= P(X=1, Y=2) /P(Y=2)= 0.05/0.45=0.111

P(X=1|Y=3)= P(X=1, Y=3) /P(Y=3)= 0.1/0.55=0.1818

Similarly for x=2,3

We see that P(X|Y) not equal to P(X). This shows that X and Y are dependent.

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