Question

Given the following joint distribution of two random variables X and Y

20.10 0.05 0.15 0.10 0.10 4 0.04 0.02 0.06 0.04 0.04 6 0.04 0.02 0.06 0.06 0.02 8 0.02 0.01 0.03 0 0.04

(a) Compute marginal distribution P_X(x)

(b) Compute marginal distribution P_Y(y)

(c) What is the conditional probability P(Y | X = 2)?

Answer 1

Definition of a marginal distribution = If X and Y are discrete random variables and f (x,y) is the value of their joint probability distribution at (x,y), the functions given by:
g(x) = Σy f (x,y) and

h(y) = Σx f (x,y) are the marginal distributions of X and Y , respectively.

X/Y	1	2	3	4	5	P(Y)	y*p(y)
2	0.1	0.05	0.15	0.1	0.1	0.5	1
4	0.04	0.02	0.06	0.04	0.04	0.2	0.8
6	0.04	0.02	0.06	0.06	0.02	0.2	1.2
8	0.02	0.01	0.03	0	0.04	0.1	0.8
P(X)	0.2	0.1	0.3	0.2	0.2
x*p(x)	0.2	0.2	0.9	0.8	1

A.

Marginal Distribution of X
X	1	2	3	4	5
P(X)	0.2	0.1	0.3	0.2	0.2

B.

Marginal Distribution of Y
Y	2	4	6	8
P(Y)	3.2	6.1	9.3	12.2

C.

P(Y | X = 2) = P(X,Y)/P(X=2)

So it would be,

Consitional probability of P(Y|X=2)
Y	2	4	6	8
P(Y|X=2)	0.5	0.2	0.2	0.1