Confirmation: getting joint cdf and marginals from joint pdf when region is 0<x<y
$begingroup$
$h(x,y)= e^{-y}$ if $0<x<y<infty$ and $=0 $ otherwise.
To get the joint cdf $H(x,y)$, that is just found by integrating $h(x,y)$ over the region ${(x,y):x leq x_0, y leq y_0, 0<x<y}$ correct?
Upon doing so, I get $H(x,y)= 1- e^{-x} - x e^{-y}$ for $0<x<y<infty$. Then to get the marginal distributions of X and Y I did the following.
$G(y)= lim_{y to infty} H(x,y)$ and $F(x)= lim_{x to y} H(x,y)$. Is the step for F(x) correct? Because x is at most y so I do not take limit to infinity? And then finally take derivative of each to get the marginal pdf's. Can anyone confirm my steps? Thanks.
probability statistics probability-distributions marginal-distribution
asked Dec 9 '18 at 23:35
jerryjerry
288
$endgroup$
add a comment |
$begingroup$
$h(x,y)= e^{-y}$ if $0<x<y<infty$ and $=0 $ otherwise.
To get the joint cdf $H(x,y)$, that is just found by integrating $h(x,y)$ over the region ${(x,y):x leq x_0, y leq y_0, 0<x<y}$ correct?
Upon doing so, I get $H(x,y)= 1- e^{-x} - x e^{-y}$ for $0<x<y<infty$. Then to get the marginal distributions of X and Y I did the following.
$G(y)= lim_{y to infty} H(x,y)$ and $F(x)= lim_{x to y} H(x,y)$. Is the step for F(x) correct? Because x is at most y so I do not take limit to infinity? And then finally take derivative of each to get the marginal pdf's. Can anyone confirm my steps? Thanks.
probability statistics probability-distributions marginal-distribution
asked Dec 9 '18 at 23:35
jerryjerry
288
$endgroup$
add a comment |
$begingroup$
$h(x,y)= e^{-y}$ if $0<x<y<infty$ and $=0 $ otherwise.
To get the joint cdf $H(x,y)$, that is just found by integrating $h(x,y)$ over the region ${(x,y):x leq x_0, y leq y_0, 0<x<y}$ correct?
Upon doing so, I get $H(x,y)= 1- e^{-x} - x e^{-y}$ for $0<x<y<infty$. Then to get the marginal distributions of X and Y I did the following.
$G(y)= lim_{y to infty} H(x,y)$ and $F(x)= lim_{x to y} H(x,y)$. Is the step for F(x) correct? Because x is at most y so I do not take limit to infinity? And then finally take derivative of each to get the marginal pdf's. Can anyone confirm my steps? Thanks.
probability statistics probability-distributions marginal-distribution
asked Dec 9 '18 at 23:35
jerryjerry
288
$endgroup$
$h(x,y)= e^{-y}$ if $0<x<y<infty$ and $=0 $ otherwise.
To get the joint cdf $H(x,y)$, that is just found by integrating $h(x,y)$ over the region ${(x,y):x leq x_0, y leq y_0, 0<x<y}$ correct?
Upon doing so, I get $H(x,y)= 1- e^{-x} - x e^{-y}$ for $0<x<y<infty$. Then to get the marginal distributions of X and Y I did the following.
$G(y)= lim_{y to infty} H(x,y)$ and $F(x)= lim_{x to y} H(x,y)$. Is the step for F(x) correct? Because x is at most y so I do not take limit to infinity? And then finally take derivative of each to get the marginal pdf's. Can anyone confirm my steps? Thanks.
probability statistics probability-distributions marginal-distribution
probability statistics probability-distributions marginal-distribution
asked Dec 9 '18 at 23:35
jerryjerry
288
asked Dec 9 '18 at 23:35
jerryjerry
288
asked Dec 9 '18 at 23:35
jerryjerry
288
asked Dec 9 '18 at 23:35
jerryjerry
288
asked Dec 9 '18 at 23:35
jerryjerry
288
288
add a comment |
add a comment |
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