Question

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.

f. Calculate E(X) and E(Y).

g. Calculate V(X) and V(Y).

h. Calculate E(X|Y=1).

Answer 1

Solution-:

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 is given in the following table:

			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 find as follows;

We know that two conditions (i) and (ii)

We preare following table :

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

(f) We find , E(X),E(Y)

  

and  

(g) We find , V(X),V(Y)

Where,

and

Where,