Question

For each of the following functions, determine the constant c so that f(x,y) satisfies the conditions of being a joint pmf for two discrete random variables X and Y:

(a) f(x,y) = c(x+2y), x=1,2, y= 1,2,3.
(b) f(x,y) = c(x+y), x=1,2,3, y=1,...,x.
(c) f(x,y) = c, x and y are integers such that 9<=x+y<=8, 0<=y<=5.
(d) f(x,y) = c((1/4)^x)((1/3)^y), x=1,2,..., y=1,2,....

Answer 1

To determine the constant c for each function so that f(x,y) satisfies the conditions of being a joint pmf for two discrete random variables X and Y, we can set up equations and solve for c.

Step-by-step explanation:

To determine the constant c for each function so that f(x,y) satisfies the conditions of being a joint pmf for two discrete random variables X and Y:

(a) For function f(x,y) = c(x+2y), we need to calculate the sum of all probabilities and set it equal to 1:

c(1+2+4+2+4+6) = 1. Solving this equation gives us c = 1/21.

(b) For function f(x,y) = c(x+y), the sum of probabilities is:

c(2+3+4+3+4+5) = 1. Solving this equation gives us c = 1/21.

(c) For function f(x,y) = c, the sum of probabilities is:

c(2+3+4+5) = 1. Solving this equation gives us c = 1/14.

(d) For function f(x,y) = c((1/4)ˣ)((1/3)y), the sum of probabilities is:

c[(1/4)+(1/4)²+...][(1/3)+(1/3)₂+...] = 1. Solving this equation gives us c = 12/121.