Fxy (x,y) = k -2<y<=2
= 0 elsewhere
where k is constant
I would like to know the value of K and I am really confused.
To find the value of k, you need to use the given function and its conditions.
The function, Fxy (x,y), is defined as follows:
Fxy (x,y) = k for -2 < y <= 2
Fxy (x,y) = 0 elsewhere
From this definition, we can determine that the value of Fxy (x,y) is only dependent on the value of y.
Let's focus on the condition -2 < y <= 2. So, for all values of x within the domain and any value of y between -2 and 2 (excluding -2 itself), Fxy (x,y) will be equal to k. Since k is a constant, it remains the same for all such x and y.
On the other hand, for all other values of y outside the range of -2 to 2 (including -2 itself), Fxy (x,y) will be equal to 0.
To find the value of k, we can use the fact that the integral of Fxy (x,y) over its whole domain must be equal to 1. This represents the normalization condition for a probability distribution.
Integrating Fxy (x,y) over its domain, we have:
∫ [ -∞ to ∞ ] ∫ [ -∞ to ∞ ] Fxy (x,y) dx dy = 1
Since Fxy (x,y) is zero everywhere except for the condition -2 < y <= 2, the integral simplifies:
∫ [ -∞ to ∞ ] ∫ [ -2 to 2 ] k dx dy = 1
The integral of a constant with respect to x is simply the constant multiplied by the integral range. Therefore:
k * [ ∫ [ -∞ to ∞ ] x dy ] from -2 to 2 = 1
k * [x * y] from -2 to 2 = 1
k * [x * 2 - x * (-2)] = 1
k * [4x] = 1
Since the integral is equal to 1, the coefficient k must be 1 divided by the integral value:
k = 1 / 4x
We can substitute any value of x here to obtain the value of k. For example, choosing x = 1:
k = 1 / 4 * 1
k = 1/4
Therefore, the value of k for the given function is 1/4.