Refer to Exercise 9.
a. Find the conditional probability mass function pYX(y4).
b. Find the conditional probability mass function pXY(x3).
c. Find the conditional expectation E(Y  X = 4).
d. Find the conditional expectation E(X  Y = 3).
Step 1 of 4:
Bolt manufactured for a certain for a certain purpose classified as acceptable,downgraded or scrap. In a lot of 500 bolts, if X denote the number of downgraded bolts, Y dente the number of scraps. The joint probability mass function of X and Y is given . here we can find the marginal probability function by adding the rows and columns.
Y 

X 
0 
1 
2 
3 
4 

0 
0.06 
0.03 
0.01 
0 
0 
0.1 
1 
0.06 
0.08 
0.04 
0.02 
0 
0.2 
2 
0.04 
0.05 
0.12 
0.06 
0.03 
0.3 
3 
0 
0.03 
0.07 
0.09 
0.06 
0.25 
4 
0 
0 
0.02 
0.06 
0.07 
0.15 
0.16 
0.19 
0.26 
0.23 
0.16 
.
We have to find
The conditional probability mass function PY/X(y/4). The conditional probability mass function PX/Y(x/3). The conditional expectation E (Y/X=4). The conditional expectation E (X/Y=3).Step 2 of 4:
The conditional probability mass function PY/X(y/4) have to find for different values of y. By using the table,
(i) PY/X(0/4) =
=
= 0
(ii) PY/X(1/4) =
=
= 0
(iii) PY/X(2/4) =
=
=
(iv) PY/X(3/4) =
=
=
(v) PY/X(4/4) =
=
=