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FRM Probability question

319.2. The following is a probability matrix for X = {1, 2, 3} and Y = {1, 2, 3}; i.e., the Joint Prob (X = 3, Y = 2) = 18.0%:
 

 
 
x

 
 
1
2
3

y
1
6%
15%
9%

2
12%
30%
18%

3
2%
5%
3%

 

Each of the following is true EXCEPT which is false?

a. X and Y are independent
b. The covariance(X,Y) is non-zero
c. The probability Y = 3 conditional on X = 1 is 10.0%; i.e., Prob (Y = 3, X = 1) = 10.0%
d. The unconditional probability that X = 2 is 50.0%; i.e., Prob (X = 2) = 50.0%

What is the reason for X and Y to be independent ?, and what if  Prob(x=1,y=1) = 0 % or 100 % 

Regards

GN

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I believe option d is wrong.

A. 1,2,3 are just data points. What is important is to define the probability function. X and Y are independent

B. Correct

C. Correct

D. It is 32 %

back against the wall. no retreat no surrender.

Rahan Nix wrote:
What is the reason for X and Y to be independent?

X and Y are independent if P(X = X0 & Y = Y0) = P(X = X0)P(Y = Y0) for any X0, Y0.

So, for example, P(X = 1) = 20% and P(Y = 3) = 10%.  Check to see if P(X = 1 & Y = 3) = 20% × 10% = 2%.

You need to check all 9 possibilities.

Perhaps easier is to check that all of the row vectors in the joint probability matrix are proportional to each other, and that all of the column vectors are proportional to each other.

Simplify the complicated side; don't complify the simplicated side.

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