Computing the joint pdf of functions of random variables using a Jacobian
$begingroup$
Let the joint pdf of $(X, Y)$ be given by
$$f(x, y) = begin{cases}
g(x+y)/(x + y) & text{ if } x > 0, y > 0 \
0 & text{ otherwise,}
end{cases} $$
where $g$ is a pdf defined on $(0, infty)$. Let $U = X/(X + Y)$ and let $V = X + Y$. Find the joint pdf of $(U, V)$.
My attempt:
Let $U = g_{1}(X, Y) = X/(X + Y)$ and $V = g_{2}(X, Y) = X + Y$. Solving for $X$ and $Y$ in terms of $U$ and $V$ yields $X = UV$ and $Y = V(1 - U)$.
Now we compute the Jacobian:
$$J =begin{vmatrix}
frac{partial g_{1}}{partial x} & frac{partial g_{1}}{partial y} \
frac{partial g_{2}}{partial x} & frac{partial g_{2}}{partial y}
end{vmatrix} = left(frac{y}{(x + y)^{2}} cdot 1right) - left(frac{1}{(x + y)^{2}} cdot 1right)$$
$$= frac{y - 1}{(x + y)^{2}} = frac{v - uv - 1}{v^{2}}$$
Therefore, the new density is given by
$$f_{UV}(u, v) = begin{cases}
v cdot g(v)/(v - uv - 1) & text{ if } u, v > 0 \
0 & text{ otherwise.}
end{cases}$$
Is this correct?
probability probability-theory proof-verification probability-distributions
asked Dec 13 '18 at 17:19
josephjoseph
503111
$endgroup$
add a comment |
$begingroup$
Let the joint pdf of $(X, Y)$ be given by
$$f(x, y) = begin{cases}
g(x+y)/(x + y) & text{ if } x > 0, y > 0 \
0 & text{ otherwise,}
end{cases} $$
where $g$ is a pdf defined on $(0, infty)$. Let $U = X/(X + Y)$ and let $V = X + Y$. Find the joint pdf of $(U, V)$.
My attempt:
Let $U = g_{1}(X, Y) = X/(X + Y)$ and $V = g_{2}(X, Y) = X + Y$. Solving for $X$ and $Y$ in terms of $U$ and $V$ yields $X = UV$ and $Y = V(1 - U)$.
Now we compute the Jacobian:
$$J =begin{vmatrix}
frac{partial g_{1}}{partial x} & frac{partial g_{1}}{partial y} \
frac{partial g_{2}}{partial x} & frac{partial g_{2}}{partial y}
end{vmatrix} = left(frac{y}{(x + y)^{2}} cdot 1right) - left(frac{1}{(x + y)^{2}} cdot 1right)$$
$$= frac{y - 1}{(x + y)^{2}} = frac{v - uv - 1}{v^{2}}$$
Therefore, the new density is given by
$$f_{UV}(u, v) = begin{cases}
v cdot g(v)/(v - uv - 1) & text{ if } u, v > 0 \
0 & text{ otherwise.}
end{cases}$$
Is this correct?
probability probability-theory proof-verification probability-distributions
asked Dec 13 '18 at 17:19
josephjoseph
503111
$endgroup$
add a comment |
$begingroup$
Let the joint pdf of $(X, Y)$ be given by
$$f(x, y) = begin{cases}
g(x+y)/(x + y) & text{ if } x > 0, y > 0 \
0 & text{ otherwise,}
end{cases} $$
where $g$ is a pdf defined on $(0, infty)$. Let $U = X/(X + Y)$ and let $V = X + Y$. Find the joint pdf of $(U, V)$.
My attempt:
Let $U = g_{1}(X, Y) = X/(X + Y)$ and $V = g_{2}(X, Y) = X + Y$. Solving for $X$ and $Y$ in terms of $U$ and $V$ yields $X = UV$ and $Y = V(1 - U)$.
Now we compute the Jacobian:
$$J =begin{vmatrix}
frac{partial g_{1}}{partial x} & frac{partial g_{1}}{partial y} \
frac{partial g_{2}}{partial x} & frac{partial g_{2}}{partial y}
end{vmatrix} = left(frac{y}{(x + y)^{2}} cdot 1right) - left(frac{1}{(x + y)^{2}} cdot 1right)$$
$$= frac{y - 1}{(x + y)^{2}} = frac{v - uv - 1}{v^{2}}$$
Therefore, the new density is given by
$$f_{UV}(u, v) = begin{cases}
v cdot g(v)/(v - uv - 1) & text{ if } u, v > 0 \
0 & text{ otherwise.}
end{cases}$$
Is this correct?
probability probability-theory proof-verification probability-distributions
asked Dec 13 '18 at 17:19
josephjoseph
503111
$endgroup$
Let the joint pdf of $(X, Y)$ be given by
$$f(x, y) = begin{cases}
g(x+y)/(x + y) & text{ if } x > 0, y > 0 \
0 & text{ otherwise,}
end{cases} $$
where $g$ is a pdf defined on $(0, infty)$. Let $U = X/(X + Y)$ and let $V = X + Y$. Find the joint pdf of $(U, V)$.
My attempt:
Let $U = g_{1}(X, Y) = X/(X + Y)$ and $V = g_{2}(X, Y) = X + Y$. Solving for $X$ and $Y$ in terms of $U$ and $V$ yields $X = UV$ and $Y = V(1 - U)$.
Now we compute the Jacobian:
$$J =begin{vmatrix}
frac{partial g_{1}}{partial x} & frac{partial g_{1}}{partial y} \
frac{partial g_{2}}{partial x} & frac{partial g_{2}}{partial y}
end{vmatrix} = left(frac{y}{(x + y)^{2}} cdot 1right) - left(frac{1}{(x + y)^{2}} cdot 1right)$$
$$= frac{y - 1}{(x + y)^{2}} = frac{v - uv - 1}{v^{2}}$$
Therefore, the new density is given by
$$f_{UV}(u, v) = begin{cases}
v cdot g(v)/(v - uv - 1) & text{ if } u, v > 0 \
0 & text{ otherwise.}
end{cases}$$
Is this correct?
probability probability-theory proof-verification probability-distributions
probability probability-theory proof-verification probability-distributions
asked Dec 13 '18 at 17:19
josephjoseph
503111
asked Dec 13 '18 at 17:19
josephjoseph
503111
asked Dec 13 '18 at 17:19
josephjoseph
503111
asked Dec 13 '18 at 17:19
josephjoseph
503111
asked Dec 13 '18 at 17:19
josephjoseph
503111
503111
add a comment |
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