Consider two ips of a coin. Let X1 be the random variable which is 1 if...

Consider two ips of a coin. Let X1 be the random variable which is 1 if the rst coin is heads and 0 otherwise; let X2 be the random variable which is 1 if the second coin is heads and 0 otherwise; and let X3 be the random variable which is 1 if the two coins are the same and 0 otherwise. Show that these three variables are pairwise independent but not independent.

Expert Solution

Proof for pairwise independence:

This means that if we consider these random variables in pairs, then the two variables are independent. For instance, consider the pair (X1,X2).

In this pair, the outcome of X1 does not affect the outcome of X2, and vice versa, because the coin is fair and is flipped one after the other. So the outcomes are completely random.

Next consider the pair (X1,X3). If you are given the value of X1, you cannot predict the value of X3, because X3 is decided based on both X1 and X2. Similarly, if you know the value of X3, then also you cannot tell the value of X1.

Let's suppose that X3 is equal to 1. This means that both coins are same, which means that X1 can either be 0 or 1. Thus there is pairwise independence in this pair.

Similar argument holds true for the pair (X2,X3) as well.

Next, consider the three variables all together. If they are all independent on individual level as well, then knowing the value of 2 of these variables should not help us in predicting the value of the remaining variable.

But this does not hold true in this case. Because if we are given the value of X1 and X2, we can tell the value of X3.

Similarly, if we are given the value of X2 and X3, or X1 and X3, then also it is possible to correctly tell the value of the remaining variable.

Thus they are only pairwise independent, but not independent as a whole.

orchestra answered 2 years ago

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