Suppose that random variables X and Y vary in accordance with the joint pdf, f_(X,Y)(x,y)=c(x+y), 0<x<y<1. Find c.

lunja55 2022-09-25 Answered
Suppose that random variables X and Y vary in accordance with the joint pdf, f X , Y ( x , y ) = c ( x + y ) , 0 < x < y < 1.. Find c.
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Answers (1)

Ashly Sanford
Answered 2022-09-26 Author has 9 answers
Given:
0 f X , Y ( x , y ) = c ( x + y )
If f X , Y is a valid pdf, then the integral over all possible values has to be equal to 1:
0 1 0 y f X , Y
( x , y ) d x d y = 0 1 0 y c ( x + y ) d x d t = c 0 1 0 y ( x + y ) d x d y
= c 0 1 ( x 2 2 + x y ) | 0 y d y = c 0 1 ( 3 2 y 2 ) d y
= c ( y 3 2 ) | 0 1 = c ( 1 2 ) = c 2
This integral has to be equal to 1:
c 2 = 1
Multiply each side by 2:
c=2
Result:
c=2
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