Need help with the following questions using Bernoulli varia

Need help with the following questions using Bernoulli variables, please:

1) Prove the following rules without the use of event notation.

2) Let X and Y be Bernoulli random variables.            

   a) Give the density function for Z

PLEASE SHOW STEPS

Thank you!

Solution

a) P(x=1 and y=1) = P(X=1)*P(y=1) = 1-P(x=0) *(1-P(Y=0)

Hence

1-P[(X=1) and (Y=1)] = P(x=0) or Y=0

b)

P[(X=1 or Y=1) and Z=1] = P(x=1,z=1)+P(y=1,z=1)

= P[(x=1 and Z=1) or (Y=1 and Z=1)]

z=(x-y)^2

x and y are binomial variables

Mean of Z = E(X^2)+E(Y^2)-2E(x)E(Y)

= npq+n^2p^2+np\'q\'+n^2p\'^2-2n^2pp\'

Z has mean as pq+np^2+np\'^2-2npp\'

P(X)P(Z) not equals P(x) P(Z) as z = x-y whole square

Hence x and z are not independent.