Question

X1 and X2 are two discrete random variables. The joint probability mass function of X1 and X2, p(x1,x2) = P(X1 = 1,X2 = x2), is given by

p(1, 1) = 0.025, p(1, 2) = 0.12, p(1, 3) = 0.21 p(2, 1) = 0.18, p(2, 2) = 0.16, p(2, 3) = c.

and p(x1, x2) = 0 otherwise.

(a) Find the value of c.

(b) Find the marginal probability mass functions of X1 and X2.

(c) Are X1 and X2 independent? Explain your answer.

(d) Find the means of X1, and X2 (E[X1] and E[X1]) using the joint probability mass

function of X1 and X2.

(e) Find the means of X1, and X2 (E[X1] and E[X1]) using the marginal probability

mass functions you found in part (b).

Answer 1

a)

Find the value of c.

b)

Find the marginal probability mass functions of X1 and X2.

Let us denote X1 by X and X2 by Y

The Marginal distribution of X and Y are given in the table.

(c) Are X and Y independent? Explain your answer.

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:

are the marginal distributions of X and Y, respectively.

If f (x, y) is the value of the joint probability distribution of the discrete random variables X and Y at (x, y), and g(x) and h(y) are the values of the marginal distributions of X at x and Y at y,
respectively, then X and Y are independent iff:

for all (x, y) within their range.

X and Y are not independent because

d)

Find the means of X1, and X2 (E[X1] and E[X1]) using the joint probability mass function of X1 and X2.

e)

Find the means of X1, and X2 (E[X1] and E[X1]) using the marginal probability mass functions you found in part (b).