Suppose that X and Y has the following joint probability distribution: x f (x,y)

Question

Suppose that X and Y has the following joint probability distribution:

x

f (x,y)

2

4

1

0.2

0.15

y

3

0.1

0.15

5

0.1

0.3

                                         

a) Find the Marginal distribution of X: g(x) => g(2) and g(4)

b) Find Marginal distribution of Y: h(y) => h(1), h(3), and h(5)

c) What is the value of f(2,1), f(4,3) and f(4,5)

d) Find P ( Y = 3 | X = 4), P ( X = 2 | Y = 3), and P ( Y = 5 | X = 2)

e) Determine whether two random variables X and Y are dependent or independent?

x

f (x,y)

2

4

1

0.2

0.15

y

3

0.1

0.15

5

0.1

0.3

Explanation / Answer

a. Marginal distribution of X:

g(2) = f(2,1) + f(2,3) + f(2,5) = 0.2+0.1+0.1 = 0.4

g(4) = f(4,1) + f(4,3) + f(4,5) = 0.15+0.15+0.3 = 0.6

b. Marginal distribution of Y:

h(1) = f(2,1) + f(4,1) = 0.2+0.15 = 0.35

h(3) = f(2,3) + f(4,3) = 0.1+0.15 = 0.25

h(5) = f(2,5) + f(4,5) = 0.1+0.3 = 0.4

c. f(2,1) = 0.2

f(4,3) = 0.15

f(4,5) = 0.3

as given in the table

d. P(Y=3 | X=4) = f(3,4) / g(4) = 0.15 / 0.60 = 0.25

P(X=2 | Y=3) = f(2,3) / h(3) = 0.1/0.25 = 0.4

P(Y=5 | X = 2) = f(2,5) / g(2) = 0.1 / 0.4 = 0.25

e. as g(2) * h(1) = 0.4*0.35 = 0.14 is not equal to f(2,1) = 0.2, X and Y are not independent, as probability marginal probabilities should be equal to joint probability for independency.

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