1. 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?

2. 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?

3. 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.

4. Re: Bernoulli Random Variable

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.