Show that the following function satisfies the proper. ties of a joint probability mass function. 1x(x-3) 1 1/4 1.5 2 1/8 1.5 3 1/4 2.5 4 1/4 3 5 1/8 Determine the following:

(a) PX 2.5.

Y <3)

(b) P(x <2.5)

(c) P(Y <3)

(d) P(X>18, Y>4.7)

(e) E(X) E(Y), F(X), and F(Y).

(f) Marginal probability distribution of the random vari- able X (g) Conditional probability distribution of Y given that X = 1.5 (h) Conditional probability distribution of X given that != 2 (1) E(X|X = 1.5) (j) Are X and Y independent?

Determine the value of c that makes the function f(x, y) = c(x + y) a joint probability mass function over the nine points with x=1,2,3 and y = 1,2,3. Determine the following:

(a) PX=1, <4)

(c) P(Y2)

(b) P(X = 1)

(d) PX <2. Y <2)

(e) E(X), E(Y), F(X), and F(F)

(f) Marginal probability distribution of the random variable.X (g) Conditional probability distribution of Y given that X = 1 (h) Conditional probability distribution of X given that Y = 2 (i) E(X= 1) (j) Are X and Y independent?