The random variable X can take on the values 1, 2 and 3 and the random variable Y can take on the values 1, 3, and 4. The joint probability distribution of X and Y is given in the following table:
Y 

1 
3 
4 

X 
1 
0.1 
0.15 
0.1 
2 
0.1 
0.1 
0.1 

3 
0.1 
0.2 
a. What value should go in the blank cell?
b. Describe in words and notation the event that has probability 0.2 in the table.
c. Calculate the marginal distribution of X and the marginal distribution of Y.
d. Are X and Y independent events? Show why or why not with calculations.
e. Calculate the conditional distribution of X given Y=1.
X and Y be the two random variable with possible values 1,2,3 and 1,3,4 respectively.
The joint probability distribution of X and Y
Y  
X\Y  1  3  4  
1  0.1  0.15  0.1  
X  2  0.1  0.1  0.1 
3  a  0.1  0.2 
(a)
We have two conditions (i) and (ii)
X\Y  1  3  4  Total 
1  0.1  0.15  0.1  0.35 
2  0.1  0.1  0.1  0.3 
3  0.05  0.1  0.2  0.35 
Total  0.25  0.35  0.4  1 
We get,
a=0.05
(b)
The event that has probability 0.2 in the table means probability of X=3 and Y=4.
i.e.
(c)
The marginal probability of X is,
X  1  2  3  Total 
P(x)  0.35  0.3  0.35  1 
The marginal probabilty of Y is,
Y  1  3  4  Total 
P(y)  0.25  0.35  0.4  1 
(d)
X and Y are called independent random variables iff ,
In other words
for all i and j.
Here,
Simalarly we find all probabilitie, condition of independecy not holds;
Here, we senen that X and Y are not independent.
(e)
The conditional distribution of X given Y=1.
Total  
Y=1  0.1  0.1  0.05  0.25 
The conditional probability of X given Y=1 ,
X  1  2  3  Total 
P(X/Y=1)  0.4  0.4  0.2  1 

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