Joint distribution function?

Question

let x, y be random variables with joint pdf

f x,y(x,y) ={ 0.25 -1<= x , y<=1

{ 0 otherwise

P (2X-Y> 0)

Can anyone help me by giving me thorough description of the region , thanks a million

Answer 1

draw the line 2x-y = 0

use some point not on the line to check if 2X-Y&gt; 0 is satisfied

for eg.(1,1) satisfies the inequality

shade that side of the line that contains (1,1)

Also note that there are other boundaries for x,y from -1 to 1 from the definition of the pdf  

Answer 2

2x -y &gt;0

2x &gt; y

x &gt; y/2

Integrate from y/2 to 1 the integral ∫ dx =1-0.5y

Then, integrate  ÃƒÂ¢Ã‚ˆÂ« (1-0.5y) dy from [-1,1]

Answer 3

the nontrivial region of the density function is a square whose vertices are (-1,-1) , (-1,1) , (1,1) &amp; (1, -1)

then we need the portion of the region below the line y = 2x .. . .

thus this would be the region with corners:

(1/2 , 1) , (1,1) , (1,-1) &amp; (-1/2 , -1) .. . .. . a trapezoid

but that would be half the region mentioned (since the initial function is homogeneous on its domain , note that the density function is constant)

thus

P(2X - Y &gt; 0) = 1/2

[as this would simply end up as being the region desired over the total domain.]