Find an example of a discrete random variable X having the f

Find an example of a discrete random variable X having the following property:

Let X₁ and X₂ be independent and identical to X. Then Z = |X₁ X₂| is also identical to X.

Solution

Let X1 and X2 iid Bernoullie (0.5) trials, P(X1=0) = P(X1=1) = P(X2=0) = P(X2=1) = 0.5

Now, P(Z=0) = P(X1=X2) = P(X1=X2=0) + P(X1=X2=1) = (0.25*0.25)+(0.25*0.25) = 0.5

And P(Z=1) = 1 - P(Z=0 ) = 1-0.5 = 0.5 (because Z can only take value 0 or 1.

Therefore Z is identical to X