Probability

Question

Let X and Y be independent random variables each taking the values -1 or 1 with probability 1/2. Let Z=XY . Show that X,Y,Z are pairwise independent .

Are they indepenedent ?

Answer 1

Yes. X, Y, Z are pairwise independent.

We'll have 4 combinations of X and Y and it's corresponding Z value will be as follows :

 X   Y     Z = XY

-1   -1       1

-1    1      -1

 1   -1      -1

 1     1      1

Now, if you consider any two pairs either XY, YZ or XZ, you can see they're independent i.e. they have all possible combinations of 1 and -1 with equal probability.

Comments

Good explanation but could you elaborate it. and could show how it is pairwise independent by applying probability rules .

---

X is a random variable which could be either 1 or -1 . It means , p(x= -1) or p(x=1) we need to find and each probability is given as 1/2 . Am i correct ?

@habibkhan

---

yes! here both probabilities are 1/2

P(X/Y) = P(X ∩ Y)/P(Y)  = P(X).P(Y) = 1/2 * 1/2 = 1/4 but we get Z = 1 for two cases

$\therefore$ P(Z) = 1/4 + 1/4 =1/2 = P(X) = P(Y)

and also $\sum P(X_{i})$ != 1

So we can conclude they are independent events