S.O.S. Mathematics CyberBoard

[Mathman]

1. If the joint probability distribution of X and Y is given by f(x, y) = c(x^2 + y^2) for x = -1, 0, 1, 3; y = -1, 2, 3, find the value of c.

With reference to the value obtained for c, find

(a) P(X <= 1, Y >2)

(b) P(X = 0, Y <=2)

(c) P(X + Y > 2)

2. If the joint probability density of X and Y is given by f(x, y) = 2 for x > 0, y > 0, x + y < 1, find

(a) P(x <= 1/2, y <= 1/2)

(b) P(X + Y > 2/3)

(c) P(X > 2Y)

[royhaas]

1. To find the value of c, use the fact that the probabilities must add to 1 for the twelve ordered pairs for which the probability is positive. Then simply add the probabilities involved in (a), (b), and (c).

2. You can do this with integrals or comparing areas. The joint density is constant over a triangular region, and in fact is the reciprocal of the area of the triangle in the first quadrant. That means that the density is uniform in the region. For example, 2a. can be computed by the ratio of the area of a square to the area of the triangle.