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I'm studying for CAS/SOA Exam 1/P and I'm stumped on this question. It says:

From the set of families with two children a family is selected at random. Let $X_1=1$ if the first child of the family is a girl; $X_2 = 1$ if the second child of the family is a girl; and $X_3 = 1$ if the family has exactly one boy. For $i=1,2,3$ let $X_i = 0$ in other cases. Determine if $X_1,X_2$, and $X_3$ are independent. Assume that in the family the probability that a child is a girl is independent of the gender of the other children and is $\frac12$.

I'm just really stuck with the events here and I think that's what's throwing me off. A little help?

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The answer may be intuitively clear if one looks at what the $X_i$ mean. One then can do a formal verification. – André Nicolas Nov 30 '12 at 5:03
b-wilson's answer is where it's at. Independence of events means that the probability of any one event happening should not change once we establish whether or not other events have happened. Each of these events has probability 1/2 individually of happening. If we have two girls, the probability that $X_3=1$ has dropped to $0$. Or from another angle (of which there are many), if we have exactly one boy ($X_3=1$) and the first child is not a girl ($X_1=0$) the probability of $X_2=1$ has risen to $1$. – alex.jordan Nov 30 '12 at 5:21
up vote 2 down vote accepted

Can $X_1$, $X_2$, and $X_3$ all be 1 at the same time?

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Does the question ask for pairwise independence or mutual? Also, good luck with your studying, actuarial exams can be daunting but absolutely rewarding!

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Thanks! It definitely is daunting since it's coming up soon. – TheHopefulActuary Nov 30 '12 at 13:09
This isn't an answer to the question. This is a comment. Please delete this answer and put it as a comment if anything. – Graphth Nov 30 '12 at 13:32

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