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June 5th, 2018, 12:00 AM   #1
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From: seoul

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How can I calculate the joint probability for three variable?

I am a student studying the joint probability density function with multi variables.

I understand how to obtain a joint probability density function when two uniform distributions have the following joint distribution like below.

The distribution $f_V$(v) can be determined based on the distribution of $t$, denoted as $f_T$(t),and that of $c_i$, denoted by $f_C$(c).

The distributions $f_T$(t) and $f_C$(c) are available, e.g., based on the previous observations. Assume that variables $t$ and $c$ are independent from each other and follow uniform distributions, i.e., $t$~ $U[t_{min}; t_{max}]$
and $c$~ $U[c_{min}; c_{max}]$, $c_{min}$ > 0. Then, $f_V$ (v) is determined
as follows.

Let, z = c, v = $\frac{t}{c}$ and we have t = vz and c = z.


Jacobian determinant $J_D$ among t, c, v and z
is given by $J_D$ = z.
So, the PDF for joint distribution $(v,z)$ is given by
$f_{V,Z}$(v,z) = $f_T$(v,z)$f_C$(z)$|J_D|$ = $\frac{1}{(t_{max} - t_{min})(c_{max}-c_{min})}|z|$.

So, the distribution of $v$, $f_V$(v) is determined by
$f_V$(v) = $\int_{-\infty}^\infty$ $f_{V,Z}$(v,z) $dz$ = $\int_{c_{min}}^{c_{max}}$ $f_{V,Z}$(v,z)$dz$ = $\frac{c_{min}+c_{max}}{2(t_{max}-t_{min})}$


The question is based on above understanding, I try to do calculations when there are 3 variables.

The distribution $f_V$(v) can be determined based on the distribution of $t$, denoted as $f_T$(t), that of $c$, denoted by $f_C$(c) and that of $L$, denoted by $f_L$(l).

The distributions $f_T$(t), $f_C$(c), and $f_L$(l) are available. Assume that variables $t$,$c$ and $l$ are independent from each other and follow uniform distributions, i.e., $t$~ $U[t_{min}; t_{max}]$
and $c$~ $U[c_{min}; c_{max}]$, $c_{min}$ > 0, $l$~ $U[l_{min}; l_{max}]$.

Then, how can i determined $f_V$ (v)? Can I get a guide for determining this joint probability function or some materials i can follow the step for this?

Let z = c, v = $\frac{t}{c*l}$ and we have t = $vzl$ and c = z.
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