For what value of c is the function
$F ( x , y ) = c x y , 0 \leq x \leq 2,0 \leq y \leq 4$ 0, elsewhere
A joint probability density function for random variables X and Y?

A)1/16
B)1/8
C)1/4
D)1/2

Question

For what value of c is the function
$F ( x , y ) = c x y , 0 \leq x \leq 2,0 \leq y \leq 4$ 0, elsewhere
A joint probability density function for random variables X and Y?

A)1/16
B)1/8
C)1/4
D)1/2

Quizplus · Accepted Answer

For $F(x, y)$ to be a joint probability density function, the integral of $F(x, y)$ over the entire range of $x$ and $y$ must equal 1. The integral is calculated as follows:$$\int_{0}^{2} \int_{0}^{4} cxy \, dy \, dx = c \int_{0}^{2} x \left( \int_{0}^{4} y \, dy \right) dx = c \int_{0}^{2} x \left[ \frac{1}{2}y^2 \right]_{0}^{4} dx = c \int_{0}^{2} x \cdot 8 \, dx = 8c \left[ \frac{1}{2}x^2 \right]_{0}^{2} = 8c \cdot 2 = 16c$$Setting this equal to 1 for the total probability gives:$$16c = 1 \Rightarrow c = \frac{1}{16}$$

For What Value of C Is the Function F(x,y)=cxy,0≤x≤2,0≤y≤4F ( x , y ) = c x y , 0 \leq x \leq 2,0 \leq y \leq 4F(x,y)=cxy,0≤x≤2,0≤y≤4

For What Value of C Is the Function $F ( x , y ) = c x y , 0 \leq x \leq 2,0 \leq y \leq 4$