Mixing
Can't understand a specific step in finding $operatorname{Cov}(X,Y)$
$begingroup$
Given the following joint PDF:
$$f_{X,Y}(t,s)=begin{cases}frac{1}{4}&,(t,s)in D \ 0 &,text{ else }end{cases}$$
where $D={(t,s):|t|+sle 2,,,sge 0}$. I need to find $operatorname{Cov}(X,Y)$.
For doing that, I started by finding the marginal density functions, so I would be able to calculate $E[X],E[Y]$.
According to the solution:
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & , 0leq s leq 1 \
0 & ,text{ else }
end{cases}
end{equation}
I got to the same one, except the region. I have no clue why $0leq s leq 1 $ is the correct region?
This is the plot:
Could anyone please explain me how this region was picked?
probability probability-distributions random-variables
edited Jan 28 at 10:37
StubbornAtom
6,30831440
asked Jan 28 at 9:52
superuser123superuser123
48628
$endgroup$
add a comment |
$begingroup$
Given the following joint PDF:
$$f_{X,Y}(t,s)=begin{cases}frac{1}{4}&,(t,s)in D \ 0 &,text{ else }end{cases}$$
where $D={(t,s):|t|+sle 2,,,sge 0}$. I need to find $operatorname{Cov}(X,Y)$.
For doing that, I started by finding the marginal density functions, so I would be able to calculate $E[X],E[Y]$.
According to the solution:
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & , 0leq s leq 1 \
0 & ,text{ else }
end{cases}
end{equation}
I got to the same one, except the region. I have no clue why $0leq s leq 1 $ is the correct region?
This is the plot:
Could anyone please explain me how this region was picked?
probability probability-distributions random-variables
edited Jan 28 at 10:37
StubbornAtom
6,30831440
asked Jan 28 at 9:52
superuser123superuser123
48628
$endgroup$
1
$begingroup$
The solution given is wrong.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 9:59
$begingroup$
Could you please elaborate?
$endgroup$
– superuser123
Jan 28 at 10:05
add a comment |
$begingroup$
Given the following joint PDF:
$$f_{X,Y}(t,s)=begin{cases}frac{1}{4}&,(t,s)in D \ 0 &,text{ else }end{cases}$$
where $D={(t,s):|t|+sle 2,,,sge 0}$. I need to find $operatorname{Cov}(X,Y)$.
For doing that, I started by finding the marginal density functions, so I would be able to calculate $E[X],E[Y]$.
According to the solution:
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & , 0leq s leq 1 \
0 & ,text{ else }
end{cases}
end{equation}
I got to the same one, except the region. I have no clue why $0leq s leq 1 $ is the correct region?
This is the plot:
Could anyone please explain me how this region was picked?
probability probability-distributions random-variables
edited Jan 28 at 10:37
StubbornAtom
6,30831440
asked Jan 28 at 9:52
superuser123superuser123
48628
$endgroup$
Given the following joint PDF:
$$f_{X,Y}(t,s)=begin{cases}frac{1}{4}&,(t,s)in D \ 0 &,text{ else }end{cases}$$
where $D={(t,s):|t|+sle 2,,,sge 0}$. I need to find $operatorname{Cov}(X,Y)$.
For doing that, I started by finding the marginal density functions, so I would be able to calculate $E[X],E[Y]$.
According to the solution:
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & , 0leq s leq 1 \
0 & ,text{ else }
end{cases}
end{equation}
I got to the same one, except the region. I have no clue why $0leq s leq 1 $ is the correct region?
This is the plot:
Could anyone please explain me how this region was picked?
probability probability-distributions random-variables
probability probability-distributions random-variables
edited Jan 28 at 10:37
StubbornAtom
6,30831440
asked Jan 28 at 9:52
superuser123superuser123
48628
edited Jan 28 at 10:37
StubbornAtom
6,30831440
asked Jan 28 at 9:52
superuser123superuser123
48628
edited Jan 28 at 10:37
StubbornAtom
6,30831440
edited Jan 28 at 10:37
StubbornAtom
6,30831440
edited Jan 28 at 10:37
StubbornAtom
6,30831440
6,30831440
asked Jan 28 at 9:52
superuser123superuser123
48628
asked Jan 28 at 9:52
superuser123superuser123
48628
asked Jan 28 at 9:52
superuser123superuser123
48628
48628
1
$begingroup$
The solution given is wrong.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 9:59
$begingroup$
Could you please elaborate?
$endgroup$
– superuser123
Jan 28 at 10:05
add a comment |
1
$begingroup$
The solution given is wrong.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 9:59
$begingroup$
Could you please elaborate?
$endgroup$
– superuser123
Jan 28 at 10:05
1
1
$begingroup$
The solution given is wrong.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 9:59
$begingroup$
The solution given is wrong.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 9:59
$begingroup$
Could you please elaborate?
$endgroup$
– superuser123
Jan 28 at 10:05
$begingroup$
Could you please elaborate?
$endgroup$
– superuser123
Jan 28 at 10:05
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
$f_Y(s)=int_{-(2-s)}^{2-s} frac 1 4 , dt= frac 12 (2-s)$ for $0leq s leq 2$.
answered Jan 28 at 10:10
Kavi Rama MurthyKavi Rama Murthy
70.6k53170
$endgroup$
add a comment |
$begingroup$
The marginal density of $Y$ is given by
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & 0leq s leq 2 \
0 & text {elsewhere}
end{cases}
end{equation}
Observe that if instead
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & 0leq s leq 1 \
0 & text {elsewhere}
end{cases}
end{equation}
then $$int_{0}^{1} f_Y (s) text {ds} = frac 3 4 neq 1.$$
and you may find that $operatorname {Cov} (X,Y) = 0$ i.e. $X$ and $Y$ are uncorrelated.
answered Jan 28 at 10:30
Dbchatto67Dbchatto67
2,184320
$endgroup$
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$f_Y(s)=int_{-(2-s)}^{2-s} frac 1 4 , dt= frac 12 (2-s)$ for $0leq s leq 2$.
answered Jan 28 at 10:10
Kavi Rama MurthyKavi Rama Murthy
70.6k53170
$endgroup$
add a comment |
$begingroup$
$f_Y(s)=int_{-(2-s)}^{2-s} frac 1 4 , dt= frac 12 (2-s)$ for $0leq s leq 2$.
answered Jan 28 at 10:10
Kavi Rama MurthyKavi Rama Murthy
70.6k53170
$endgroup$
add a comment |
$begingroup$
$f_Y(s)=int_{-(2-s)}^{2-s} frac 1 4 , dt= frac 12 (2-s)$ for $0leq s leq 2$.
answered Jan 28 at 10:10
Kavi Rama MurthyKavi Rama Murthy
70.6k53170
$endgroup$
$f_Y(s)=int_{-(2-s)}^{2-s} frac 1 4 , dt= frac 12 (2-s)$ for $0leq s leq 2$.
answered Jan 28 at 10:10
Kavi Rama MurthyKavi Rama Murthy
70.6k53170
answered Jan 28 at 10:10
Kavi Rama MurthyKavi Rama Murthy
70.6k53170
answered Jan 28 at 10:10
Kavi Rama MurthyKavi Rama Murthy
70.6k53170
answered Jan 28 at 10:10
Kavi Rama MurthyKavi Rama Murthy
70.6k53170
70.6k53170
add a comment |
add a comment |
$begingroup$
The marginal density of $Y$ is given by
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & 0leq s leq 2 \
0 & text {elsewhere}
end{cases}
end{equation}
Observe that if instead
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & 0leq s leq 1 \
0 & text {elsewhere}
end{cases}
end{equation}
then $$int_{0}^{1} f_Y (s) text {ds} = frac 3 4 neq 1.$$
and you may find that $operatorname {Cov} (X,Y) = 0$ i.e. $X$ and $Y$ are uncorrelated.
answered Jan 28 at 10:30
Dbchatto67Dbchatto67
2,184320
$endgroup$
add a comment |
$begingroup$
The marginal density of $Y$ is given by
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & 0leq s leq 2 \
0 & text {elsewhere}
end{cases}
end{equation}
Observe that if instead
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & 0leq s leq 1 \
0 & text {elsewhere}
end{cases}
end{equation}
then $$int_{0}^{1} f_Y (s) text {ds} = frac 3 4 neq 1.$$
and you may find that $operatorname {Cov} (X,Y) = 0$ i.e. $X$ and $Y$ are uncorrelated.
answered Jan 28 at 10:30
Dbchatto67Dbchatto67
2,184320
$endgroup$
add a comment |
$begingroup$
The marginal density of $Y$ is given by
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & 0leq s leq 2 \
0 & text {elsewhere}
end{cases}
end{equation}
Observe that if instead
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & 0leq s leq 1 \
0 & text {elsewhere}
end{cases}
end{equation}
then $$int_{0}^{1} f_Y (s) text {ds} = frac 3 4 neq 1.$$
and you may find that $operatorname {Cov} (X,Y) = 0$ i.e. $X$ and $Y$ are uncorrelated.
answered Jan 28 at 10:30
Dbchatto67Dbchatto67
2,184320
$endgroup$
The marginal density of $Y$ is given by
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & 0leq s leq 2 \
0 & text {elsewhere}
end{cases}
end{equation}
Observe that if instead
begin{equation}
f_{Y}(s)=
begin{cases}
frac{1}{2}(2-s) & 0leq s leq 1 \
0 & text {elsewhere}
end{cases}
end{equation}
then $$int_{0}^{1} f_Y (s) text {ds} = frac 3 4 neq 1.$$
and you may find that $operatorname {Cov} (X,Y) = 0$ i.e. $X$ and $Y$ are uncorrelated.
answered Jan 28 at 10:30
Dbchatto67Dbchatto67
2,184320
edited Jan 28 at 14:47
answered Jan 28 at 10:30
Dbchatto67Dbchatto67
2,184320
answered Jan 28 at 10:30
Dbchatto67Dbchatto67
2,184320
answered Jan 28 at 10:30
Dbchatto67Dbchatto67
2,184320
2,184320
add a comment |
add a comment |
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1
$begingroup$
The solution given is wrong.
$endgroup$
– Kavi Rama Murthy
Jan 28 at 9:59
$begingroup$
Could you please elaborate?
$endgroup$
– superuser123
Jan 28 at 10:05