To find the value of K, we can use one of the given pairs of (x, y) values.
Given x = -2 and y = 3, we can substitute these values into the equation:
(x, y) = (x + 2y) / K
(-2, 3) = (-2 + 2(3)) / K
(-2, 3) = (-2 + 6) / K
(-2, 3) = 4 / K
To find K, we can rearrange the equation:
4 = (-2, 3) * K
K = 4 / (-2, 3)
Therefore, the value of K is -2/3.
b. The marginal function of x:
To find the marginal function of x, we need to sum the joint probabilities over all possible y values for each x value.
For x = -2:
f(-2) = f(-2, 3) + f(-2, 4)
For x = 1:
f(1) = f(1, 3) + f(1, 4)
c. The marginal function of y:
To find the marginal function of y, we need to sum the joint probabilities over all possible x values for each y value.
For y = 3:
f(3) = f(-2, 3) + f(1, 3)
For y = 4:
f(4) = f(-2, 4) + f(1, 4)
d. To find f(x|y = 4), we can use the joint probability distribution function:
f(x|y = 4) = f(x, y) / f(y = 4)
We can substitute the values into the equation and calculate the probabilities based on the given joint probability distribution function.