Proving Independence using joint pdf












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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:




  1. The constant c

  2. The marginal pdf of X and Y

  3. Are the two random variables independent? why?

  4. 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?










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  • $begingroup$
    Yes, that seems right to me. Welcome to stack exchange. :)
    $endgroup$
    – Delsilon
    Dec 17 '18 at 16:47












  • $begingroup$
    @Delsilon thank you very much
    $endgroup$
    – zolman
    Dec 17 '18 at 16:55
















0












$begingroup$


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:




  1. The constant c

  2. The marginal pdf of X and Y

  3. Are the two random variables independent? why?

  4. 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?










share|cite|improve this question











$endgroup$












  • $begingroup$
    Yes, that seems right to me. Welcome to stack exchange. :)
    $endgroup$
    – Delsilon
    Dec 17 '18 at 16:47












  • $begingroup$
    @Delsilon thank you very much
    $endgroup$
    – zolman
    Dec 17 '18 at 16:55














0












0








0





$begingroup$


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:




  1. The constant c

  2. The marginal pdf of X and Y

  3. Are the two random variables independent? why?

  4. 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?










share|cite|improve this question











$endgroup$




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:




  1. The constant c

  2. The marginal pdf of X and Y

  3. Are the two random variables independent? why?

  4. 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?







probability probability-distributions






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edited Dec 17 '18 at 16:47









Delsilon

15311




15311










asked Dec 17 '18 at 16:27









zolmanzolman

34




34












  • $begingroup$
    Yes, that seems right to me. Welcome to stack exchange. :)
    $endgroup$
    – Delsilon
    Dec 17 '18 at 16:47












  • $begingroup$
    @Delsilon thank you very much
    $endgroup$
    – zolman
    Dec 17 '18 at 16:55


















  • $begingroup$
    Yes, that seems right to me. Welcome to stack exchange. :)
    $endgroup$
    – Delsilon
    Dec 17 '18 at 16:47












  • $begingroup$
    @Delsilon thank you very much
    $endgroup$
    – zolman
    Dec 17 '18 at 16:55
















$begingroup$
Yes, that seems right to me. Welcome to stack exchange. :)
$endgroup$
– Delsilon
Dec 17 '18 at 16:47






$begingroup$
Yes, that seems right to me. Welcome to stack exchange. :)
$endgroup$
– Delsilon
Dec 17 '18 at 16:47














$begingroup$
@Delsilon thank you very much
$endgroup$
– zolman
Dec 17 '18 at 16:55




$begingroup$
@Delsilon thank you very much
$endgroup$
– zolman
Dec 17 '18 at 16:55










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