Proof of a two-dimensional random variable
$begingroup$
A two-dimensional random variable $ (X,Y) $ has total probability density function like $ f _{X,Y} (x,y) $.
And I want to prove that probability density function of a random variable $Z=X+Y$
is $f_{Z}(z) = int_{- infty }^{ infty } f _{X,Y} (x, z-x)dx$ .
And I don't know how to do it? Any help.
probability
edited Dec 28 '18 at 9:46
Hayk
2,7671215
asked Dec 28 '18 at 8:51
AbakusAbakus
32
$endgroup$
add a comment |
$begingroup$
A two-dimensional random variable $ (X,Y) $ has total probability density function like $ f _{X,Y} (x,y) $.
And I want to prove that probability density function of a random variable $Z=X+Y$
is $f_{Z}(z) = int_{- infty }^{ infty } f _{X,Y} (x, z-x)dx$ .
And I don't know how to do it? Any help.
probability
edited Dec 28 '18 at 9:46
Hayk
2,7671215
asked Dec 28 '18 at 8:51
AbakusAbakus
32
$endgroup$
1
$begingroup$
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
$endgroup$
– Michael
Dec 28 '18 at 8:57
$begingroup$
Apply Fubini's Theorem.
$endgroup$
– Kavi Rama Murthy
Dec 28 '18 at 9:26
add a comment |
$begingroup$
A two-dimensional random variable $ (X,Y) $ has total probability density function like $ f _{X,Y} (x,y) $.
And I want to prove that probability density function of a random variable $Z=X+Y$
is $f_{Z}(z) = int_{- infty }^{ infty } f _{X,Y} (x, z-x)dx$ .
And I don't know how to do it? Any help.
probability
edited Dec 28 '18 at 9:46
Hayk
2,7671215
asked Dec 28 '18 at 8:51
AbakusAbakus
32
$endgroup$
A two-dimensional random variable $ (X,Y) $ has total probability density function like $ f _{X,Y} (x,y) $.
And I want to prove that probability density function of a random variable $Z=X+Y$
is $f_{Z}(z) = int_{- infty }^{ infty } f _{X,Y} (x, z-x)dx$ .
And I don't know how to do it? Any help.
probability
probability
edited Dec 28 '18 at 9:46
Hayk
2,7671215
asked Dec 28 '18 at 8:51
AbakusAbakus
32
edited Dec 28 '18 at 9:46
Hayk
2,7671215
asked Dec 28 '18 at 8:51
AbakusAbakus
32
edited Dec 28 '18 at 9:46
Hayk
2,7671215
edited Dec 28 '18 at 9:46
Hayk
2,7671215
edited Dec 28 '18 at 9:46
Hayk
2,7671215
2,7671215
asked Dec 28 '18 at 8:51
AbakusAbakus
32
asked Dec 28 '18 at 8:51
AbakusAbakus
32
asked Dec 28 '18 at 8:51
AbakusAbakus
32
32
1
$begingroup$
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
$endgroup$
– Michael
Dec 28 '18 at 8:57
$begingroup$
Apply Fubini's Theorem.
$endgroup$
– Kavi Rama Murthy
Dec 28 '18 at 9:26
add a comment |
1
$begingroup$
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
$endgroup$
– Michael
Dec 28 '18 at 8:57
$begingroup$
Apply Fubini's Theorem.
$endgroup$
– Kavi Rama Murthy
Dec 28 '18 at 9:26
1
1
$begingroup$
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
$endgroup$
– Michael
Dec 28 '18 at 8:57
$begingroup$
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
$endgroup$
– Michael
Dec 28 '18 at 8:57
$begingroup$
Apply Fubini's Theorem.
$endgroup$
– Kavi Rama Murthy
Dec 28 '18 at 9:26
$begingroup$
Apply Fubini's Theorem.
$endgroup$
– Kavi Rama Murthy
Dec 28 '18 at 9:26
add a comment |
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1
$begingroup$
Start with $P[Z leq z] =...$ and differentiate with respect to $z$. You can either do a 2-d integral to compute $P[Zleq z]$, or teh law of total probability.
$endgroup$
– Michael
Dec 28 '18 at 8:57
$begingroup$
Apply Fubini's Theorem.
$endgroup$
– Kavi Rama Murthy
Dec 28 '18 at 9:26