Let X and Y be Bernoulli variables PX1r PY1X0p0 ans PY1X1p1

Let X and Y be Bernoulli variables,

P(X=1)=r
P(Y=1|X=0)=p0
ans P(Y=1|X=1)=p1

where p0 is not equal to p1
define Z=(2-X)(1-Y),

Assume:

0<p<1
0<p0<1
and 0<p1<1

a) Find E(Z)

Solution

x

0

1

y

0

1

z

1

0

prob

1-p1

p1

Var(z)=E()-

E()==1-p1

==1-p1

Var(z)=E()-=1-p1-=p1(1-p1)