Conditional distribution between jointly normal and univariate normal random variables
$begingroup$
Let $X = (X_1, X_2, X_3)^Tsim N_3(mu, Sigma)$, where
$$mu = begin{pmatrix}
1 \
1 \
1 \
end{pmatrix}, Sigma = begin{pmatrix}
1 & 1 & 0 \
1 & 2 & 1 \
0 & 1 & 2 \
end{pmatrix} $$
I want to find $(X_1, X_2)^T|X_3$.
I know that $(X_1, X_2)^T sim N_2(begin{pmatrix}
1\
1 \
end{pmatrix}, begin{pmatrix}
1 & 1 \
1 & 2 \
end{pmatrix})$ and that $X_3 sim N_1(1, 2)$.
My definition of conditional distribution of MVN however initially requires the distribution of $((X_1, X_2)^T, X_3)$, of which I am not sure.
Is there another way around this that I am missing? Thanks for your help.
probability-distributions random-variables normal-distribution conditional-probability
asked Dec 23 '18 at 12:23
JasonJason
82
$endgroup$
add a comment |
$begingroup$
Let $X = (X_1, X_2, X_3)^Tsim N_3(mu, Sigma)$, where
$$mu = begin{pmatrix}
1 \
1 \
1 \
end{pmatrix}, Sigma = begin{pmatrix}
1 & 1 & 0 \
1 & 2 & 1 \
0 & 1 & 2 \
end{pmatrix} $$
I want to find $(X_1, X_2)^T|X_3$.
I know that $(X_1, X_2)^T sim N_2(begin{pmatrix}
1\
1 \
end{pmatrix}, begin{pmatrix}
1 & 1 \
1 & 2 \
end{pmatrix})$ and that $X_3 sim N_1(1, 2)$.
My definition of conditional distribution of MVN however initially requires the distribution of $((X_1, X_2)^T, X_3)$, of which I am not sure.
Is there another way around this that I am missing? Thanks for your help.
probability-distributions random-variables normal-distribution conditional-probability
asked Dec 23 '18 at 12:23
JasonJason
82
$endgroup$
$begingroup$
Cross posted at stats.stackexchange.com/q/384280/119261.
$endgroup$
– StubbornAtom
Dec 24 '18 at 5:51
add a comment |
$begingroup$
Let $X = (X_1, X_2, X_3)^Tsim N_3(mu, Sigma)$, where
$$mu = begin{pmatrix}
1 \
1 \
1 \
end{pmatrix}, Sigma = begin{pmatrix}
1 & 1 & 0 \
1 & 2 & 1 \
0 & 1 & 2 \
end{pmatrix} $$
I want to find $(X_1, X_2)^T|X_3$.
I know that $(X_1, X_2)^T sim N_2(begin{pmatrix}
1\
1 \
end{pmatrix}, begin{pmatrix}
1 & 1 \
1 & 2 \
end{pmatrix})$ and that $X_3 sim N_1(1, 2)$.
My definition of conditional distribution of MVN however initially requires the distribution of $((X_1, X_2)^T, X_3)$, of which I am not sure.
Is there another way around this that I am missing? Thanks for your help.
probability-distributions random-variables normal-distribution conditional-probability
asked Dec 23 '18 at 12:23
JasonJason
82
$endgroup$
Let $X = (X_1, X_2, X_3)^Tsim N_3(mu, Sigma)$, where
$$mu = begin{pmatrix}
1 \
1 \
1 \
end{pmatrix}, Sigma = begin{pmatrix}
1 & 1 & 0 \
1 & 2 & 1 \
0 & 1 & 2 \
end{pmatrix} $$
I want to find $(X_1, X_2)^T|X_3$.
I know that $(X_1, X_2)^T sim N_2(begin{pmatrix}
1\
1 \
end{pmatrix}, begin{pmatrix}
1 & 1 \
1 & 2 \
end{pmatrix})$ and that $X_3 sim N_1(1, 2)$.
My definition of conditional distribution of MVN however initially requires the distribution of $((X_1, X_2)^T, X_3)$, of which I am not sure.
Is there another way around this that I am missing? Thanks for your help.
probability-distributions random-variables normal-distribution conditional-probability
probability-distributions random-variables normal-distribution conditional-probability
asked Dec 23 '18 at 12:23
JasonJason
82
asked Dec 23 '18 at 12:23
JasonJason
82
asked Dec 23 '18 at 12:23
JasonJason
82
asked Dec 23 '18 at 12:23
JasonJason
82
asked Dec 23 '18 at 12:23
JasonJason
82
82
$begingroup$
Cross posted at stats.stackexchange.com/q/384280/119261.
$endgroup$
– StubbornAtom
Dec 24 '18 at 5:51
add a comment |
$begingroup$
Cross posted at stats.stackexchange.com/q/384280/119261.
$endgroup$
– StubbornAtom
Dec 24 '18 at 5:51
$begingroup$
Cross posted at stats.stackexchange.com/q/384280/119261.
$endgroup$
– StubbornAtom
Dec 24 '18 at 5:51
$begingroup$
Cross posted at stats.stackexchange.com/q/384280/119261.
$endgroup$
– StubbornAtom
Dec 24 '18 at 5:51
add a comment |
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$begingroup$
Cross posted at stats.stackexchange.com/q/384280/119261.
$endgroup$
– StubbornAtom
Dec 24 '18 at 5:51