Thread: Bernoulli Random Variable

Going over past papers, and i cant seem to interpret this question properly, and dont know if my answer is correct;


Suppose random variables X and Y are independent and each had a bernoulli distribution with parameter p = 0.5. Define Z= | X - Y |

(i) Write down all the possible values of (X,Y) and the corresponding values of Z. Hence verify that Z also has a bernoulli distribution with parameter p = 0.5

Im unsure of what its asking for.

I know that X(success) = 1, X(failure) = 0, Y(success) = 1, Y(failure) = 0

Therefore this would mean values of Z = 0,1 with the probability of each being 1/2, therefore this means parameter p for z = 0.5

Is this correct so far?

x = 0, y = 0, x-y = 0
x = 0, y = 1, x-y = -1
x = 1, y = 0, x-y = 1
x = 1, y = 1, x-y = 0

Are there other possible outcomes?

You should write all all values of the random vector (X,Y), i.e. (0,0), (0,1), (1,0), (1,1), their corresponding probabilities, and the corresponding value of Z.

X = 0 Y = 0 Z = 0 P = 1/4
X = 0 Y = 1 Z = 1 P = 1/4
X = 1 Y = 0 Z = 1 P = 1/4
X = 1 Y = 1 Z = 0 P = 1/4

Therefore, Z can two values: 0 and 1 with probability of success 1/2.
This is again a Bernoulli.

But, what about Z = X - Y?

Originally Posted by ilanshom

You should write all all values of the random vector (X,Y), i.e. (0,0), (0,1), (1,0), (1,1), their corresponding probabilities, and the corresponding value of Z.