Question

You were given the following joint probability function for

Y1 = { 0, if the child survived, 1, if not,

and

Y2 = { 0 if no belt used, 1 if adult belt used, and 2 if car seat belt used


Notice that Y1 is the number of fatalities per child and, since children's car seats usually utilize two belts, Y2 is the number of seat belts in use at the time of accident.


Given:
Y1
y2 0 1 total
0 0.38 0.17 0.55
1 0.14 0.02 0.16
2 0.24 0.05 0.29
Total 0.76 0.24 1


Are Y1 and Y2 independent? Why or why not?

Answer 1

Answer: No, Y1 and Y2 are not independent

Explanation:

Because they don't satisfy this condition:

FsubscriptY1Y2(Y1,Y2) = FsubscriptY1(y1) × FsubscriptY2(y2)

... for all given values of Y1 and Y2

This is the condition for independence.

How do we know that Y1 and Y2 don't satisfy this condition?

We use the information in the Joint Probability Distribution Table.

Let's see if the condition stands when Y1 is zero and Y2 is zero

FsubscriptY1Y2(0,0) = 0.38

FsubscriptY1(0) × FsubscriptY2(0) = 0.76×0.55 = 0.418

We can see that 0.38 is not equal to 0.418

Doing the test for any other combination of Y1 and Y2 values will give unequal figures as well.