Question

X and Y are discrete random variables with joint distribution given below Y 1 Y 0 Y 1 X 1 0 1/4 0

X 0 1/4 1/4 1/4

(a) Determine the conditional expectation E Y|X 1 . (b) Determine the conditional expectation E X|Y y for each value of y. (c) Determine the expected value of X using conditional expectation results form part (b)

above. (d) Now obatin the marginal distribution of X and verify your answers to part (c).

Answer 1

Answer:-

Given that:-

X and Y are discete random variables with joint distribution

	Y=-1	Y=0	Y=1	Total
X=1	0	1/4	0	1/4
X=0	1/4	1/4	1/4	3/4
Total	1/4	2/4	1/4	1

a) Determine condition Expectation E(y/x=1)

E(y/x=1)=

Condition probability distribution of y given x=1

Condition probability distribution of y/x=1

y	-1	0	1
p(y/x=1)	0	1	0	1

b)Detremine the condition expectation E(x/y=y) for each value or y

E(x/y=-1), E(x/y=0), E(x/y=1)

	Y=-1	Y=0	Y=1	Total
X=1	0	1/4	0	1/4
X=0	1/4	1/4	1/4	3/4
Total	1/4	2/4	1/4	1

To find E(x/y=-1)

Condition prob.distribution of x/y=-1

Condition prob distribution of x/y=-1

x	1	0
p(x/y=-1)	0	1	1

E(x/y=-1)=

E(x/y=-1)=0

To find E(x/y=0)

condition probability of x/y=0

condition prob distribution of x/y=0

x	1	0
p(x/y=0)	1/2	1/2	1

E(x/y=0)=

To find E(x/y=1)

Condition probability of x/y=1

condition prob distribution or x/y=1

x	1	0
p(x/y=1)	0	1	1

E(x/y=1)=

c)We know that from low of it created

expectation

E[E(x/y)]=E(x)

g(y)=E(x/y=y)

E[E(x/y)]=E(g(y))

=

  

E[E(x/y=y)]=E(x/y=-1).p(y=-1)+E(x/y=0)(y=0)+E(x/y=1).p(y=1)

E[E(x/y)]=

Therefore E(x)=

Therefore, Expected value of x using condition expectation from b]

d)To obtain marginal distribution of x.

	Y=-1	Y=0	Y=1	Total
X=1	0	1/4	0	1/4
X=0	1/4	1/4	1/4	3/4
Total	1/4	2/4	1/4	1

Marginal distribution of x

x	1	0
p(x)	1/4	3/4	1

E(x)=

Now from (c) and (d) we verify that