A certain compound that is used in forming a composite structural materiel depends on the interaction of two materials X & Y the percentage of having material X and Y within the composite varies from 15% to 35% and 20% to 35%, respectively however a sample is considered to be failed if the existence of each X and Y exceeds 30% of the overall composed suppose that the joint density function of these random variables is
$f(x,y)$ = \begin{cases} c(x+2y) & 0.15<x< 0.35, 0.2<y<0.35 \\ 0 & otherwise \end{cases}
Determine:
- The constant c
- The marginal pdf of X and Y
- Are the two random variables independent? why?
- The probability of having a failed sample.
My questions are about question 3 and 4
For question 3 if the combination of $f_X(x)$ and $f_Y(y)$ we got from question 2 not equal to the $f(x,y)$ then random variables are dependent, is that right?
and for number 4 we do integration twice of $f(x,y)$ in regards to $dx,dy$ and the limits of integration will be from 0.3 to 0.35 for both integration? is that correct and are the integration limits correct?