Refer to Exercise 31. Assume the first card is not replaced before the second card is drawn.
a. Find the joint probability mass function of X and Y.
b. Find the marginal probability mass functions pX(x) and pY(y).
c. Find μx and μy.
d. Find μXY.
e. Find Cov(X, Y).
Answer
Step 1 of 5</p>
a) For constructing joint probability mass function
First we need to write the range of X and y
Here X represents the number on the first card
Then X=1,2,3
Here Y represents the number on the first card
Then Y=1,2,3
The first card is not replaced before second card is drawn
So the probability of getting the same card is 0
The remaining possibilities are (1,2),(1,3),(2,1),(2,3),(3,1),(3,2)
The number of outcomes are 6
Each outcome having the equal probability=⅙
Now we are constructing the joint probability mass function
Y\ X 
1 
2 
3 

1 
0 
1/6 
1/6 
1/3 
2 
1/6 
0 
1/6 
1/3 
3 
1/6 
1/6 
0 
1/3 
1/3 
1/3 
1/3 
1 
Step 2 of 5</p>
b) The marginal probability mass function of X is
X 
1 
2 
3 
Total 
1/3 
1/3 
1/3 
1 
The marginal probability mass function of Y is
Y 
1 
2 
3 
Total 
1/3 
1/3 
1/3 
1 
Step 3 of 5</p>
c) Mean(=E(X)
=
= 1(⅓)+2(⅓)+3(⅓)
= 2
Mean(=E(Y)
=
= 1(⅓)+2(⅓)+3(⅓)
= 2