X_{1}, X_{2} is a random sample from Bernoulli distribution f(x)=p^{x} (1-p)^{(1-x)} where x=0,1

A joint distribution is defined as P(x = X, y = Y). Here, our x and y are X1 and X2 respectively, while X and Y are 0 and 1. Therefore, we're looking for this answer:

P(x = 0, y = 0) = ?

P(x = 1, y = 0) = ?

P(x = 0, y = 1) = ?

P(x = 1, y = 1) = ?

What exactly is the probability that x (or y) is equal to zero (just one number)? We plug the number in to the distribution function:

f(0)=p^{0} (1-p)^{(1-0)} = (1)(1-p) = 1-p

What about x or y being equal to one?

f(1)=p^{1} (1-p)^{(1-1)} = (p)(1-p)^{0} = p

Therefore, we have our answer as follows:

P(x = 0, y = 0) = (1-p)(1-p) = (1-p)^{2}

P(x = 1, y = 0) = p(1-p)

P(x = 0, y = 1) = (1-p)p = p(1-p)

P(x = 1, y = 1) = (p)(p) = p^{2}