Question

Let \({\bf{f}}\left( {{{\bf{x}}_{{\bf{1}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right){\bf{ = cg}}\left( {{{\bf{x}}_{{\bf{1}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right)\) be a joint p.d.f for \(\left( {{{\bf{x}}_{{\bf{1}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right){\bf{,}}\)each \({x_{2\,}}\)let\({{\bf{h}}_{{\bf{2}}\,}}\left( {{{\bf{x}}_{{\bf{1}}\,}}} \right){\bf{ = g}}\left( {{{\bf{x}}_{{\bf{1}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right)\) that is what we get by considering \({\bf{g}}\left( {{{\bf{x}}_{{\bf{1}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right)\) as a function of \({{\bf{x}}_{{\bf{1}}\,}}\)for fixed \({{\bf{x}}_{2\,}}\)show that there is a multiplicative factor \({{\bf{c}}_{{\bf{2}}\,}}\)that does not depend on such that is the conditional p.d.f of \({{\bf{x}}_{{\bf{1}}\,}}\) given \(\left( {{{\bf{x}}_{{\bf{2}}\,}}{\bf{,}}{{\bf{x}}_{{\bf{2}}\,}}} \right)\)

Short answer

That is what we get by considering.

\({g_1}\left( {{x_1}\mid {x_2}} \right)\frac{{\left( {{x_1},{x_2}} \right)}}{{{f_2}\left( {{x_2}} \right)\,\,\,\,}} = \frac{{cg\left( {{x_1},{x_2}} \right)}}{{{f_2}\left( {{x_2}} \right)\,\,\,\,}}.\)

\({x_2}\,\,\,\)is fixed, and for each \({x_2}\,\,\,\)

\({h_2}\left( {{x_1}} \right)\, = g\left( {{x_1},{x_2}} \right),\)

\({c_2} = \frac{c}{{{f_2}\left( {{x_2}} \right)\,}}.\)

Step-by-step solution

Definition of the conditional probability density function

The random variable's probability distribution is updated by taking into account some information that gives rise to a conditional probability distribution. A conditional probability density function can characterize this distribution.

The conditional probability density function of \({X_2}\,\), given that \({X_2} = {x_2}\,\,\)

\({g_1}\left( {{x_1}\mid {x_2}} \right)\frac{{\left( {{x_1},{x_2}} \right)}}{{{f_2}\left( {{x_2}} \right)\,\,\,\,}} = \frac{{cg\left( {{x_1},{x_2}} \right)}}{{{f_2}\left( {{x_2}} \right)\,\,\,\,}}.\)

\({x_2}\,\,\,\)is fixed, and for each \({x_2}\,\,\,\)

\({h_2}\left( {{x_1}} \right)\, = g\left( {{x_1},{x_2}} \right),\)

The conditional probability density function of \({X_2}\,\),given that \({X_2} = {x_2}\,\,\,\,\)

Conditional probability

A conditional probability density function can characterize this distribution.

Then the conditional probability density function of \({X_2}\,\)is

given that \({X_2} = {x_2}\,\,\,\,\)

\({g_1}\left( {{x_1}\mid {x_2}} \right)\,\, = \frac{{cg\left( {{x_1},{x_2}} \right)}}{{{f_2}\left( {{x_2}} \right)\,\,\,\,}}{h_2}\left( {{x_1}} \right)\, = {c_2}\)

\({c_2} = \frac{c}{{{f_2}\left( {{x_2}} \right)\,}}.\)

hence,

\({c_2} = \frac{c}{{{f_2}\left( {{x_2}} \right)\,}}.\)