Question

Suppose that\({X_1}....{X_n}\)are i.i.d. random variables, each having the following c.d.f.:\(F\left( x \right) = \left\{ \begin{array}{l}0\,\,\,\,\,\,\,\,\,\,\,\,\,\,for\,x \le 0\\1 - {e^{ - x}}\,\,\,for\,x > 0\end{array} \right.\)

Let\({Y_1} = min\left\{ {{X_1},{X_2}..{X_n}} \right\}\)and\({Y_n} = max\left\{ {{X_{1,}}{X_2}..{X_n}} \right\}\)Determine the conditional p.d.f. of\({Y_1}\)given that\({Y_n} = {y_n}\)

Short answer

Conditional pdf of\({Y_1}\)given that\({Y_n} = {y_n}\)is

\(h\left( {{y_1}|{y_n}} \right) = \frac{{\left( {n - 1} \right)\left( {\exp \left( { - {y_1}} \right) - \exp {{\left( { - {y_n}} \right)}^{n - 2}}\exp \left( { - {y_1}} \right)} \right)}}{{{{\left( {1 - \exp \left( { - {y_n}} \right)} \right)}^{n - 1}}}}\) for \(0 < {y_1} < {y_n}\)

Step-by-step solution

Given information

\({X_1}....{X_n}\)are iid random variables with pdf

Calculating Conditional pdf

Here,

\(f\left( x \right) = \frac{{dF\left( x \right)}}{{dx}}\)\(\)

\( = {{\mathop{\rm e}\nolimits} ^{ - x}}\)\({\rm{for}}\,x > 0\)

Hence

\(g\left( {{y_{1,}}{y_n}} \right) = n\left( {n - 1} \right){\left( {\exp \left( { - {y_1}} \right) - \exp \left( { - {y_n}} \right)} \right)^{n - 2}}\exp \left( { - \left( {{y_1} + {y_n}} \right)} \right)\)

For\(0 < {y_1} < {y_n}\)

Marginal pdf of\({Y_n}\):

\({g_n}\left( {{y_n}} \right) = n{\left( {1 - \exp \left( { - {y_n}} \right)} \right)^{n - 1}}\exp \left( { - {y_n}} \right)\)for\({y_n} > 0\)

Hence, Conditional pdf is

\(h\left( {{y_1}|{y_n}} \right) = \frac{{\left( {n - 1} \right)\left( {\exp \left( { - {y_1}} \right) - \exp {{\left( { - {y_n}} \right)}^{n - 2}}\exp \left( { - {y_1}} \right)} \right)}}{{{{\left( {1 - \exp \left( { - {y_n}} \right)} \right)}^{n - 1}}}}\) for \(0 < {y_1} < {y_n}\)