91Ó°ÊÓ

\({X_1},{X_2}\)and\({X_3}\)forms a random sample of three observations

The pdr of the distribution is\(f\left( x \right) = \left\{ \begin{array}{l}2x\,\,for\,0 < x < 1\\0\,\,otherwise\end{array} \right.\)

Let \(W = {Y_n} - {Y_1}\) and \(Z = {Y_1}\)

The joint pdf\(g\left( {w,z} \right)\)of\(\left( {W,Z} \right)\)for\(0 < w < 1\)and\(0 < z < 1 - w\):

\(\begin{aligned}{}g\left( {w,z} \right) &= 24\left[ {{{\left( {w + z} \right)}^2} - {z^2}} \right]z\left( {w + z} \right)\\ &= 24w\left( {2{z^3} + 3w{z^2} + {w^2}z} \right)\end{aligned}\)

Hence

\(g\left( {w,z} \right) = \left\{ \begin{array}{l}24w\left( {2{z^3} + 3w{z^2} + {w^2}z} \right)\,for0 < w < 1\,and0 < z < 1 - w\\0\,\,otherwise\end{array} \right.\)

Hence the pdf range for 0<w<1:

\(h\left( w \right) = \int\limits_0^{1 - w} {g\left( {w,z} \right)dz} \)

therefore

\(h\left( w \right) = 12w{\left( {1 - w} \right)^2}\)

Hence Pdf of range of the sample is \(h\left( w \right) = 12w{\left( {1 - w} \right)^2}\)