1.5 Joint Distributions

Let \(X_1,X_2,..........,X_n\) be a random sample from exponential with mean \(1/\lambda \). Let us make the following transformation \[Y_1=X_1,\hspace {0.2cm} Y_2=X_1+X_2,.....,\hspace {0.2cm} Y_j=X_1+X_2+....+X_j,.......,\hspace {0.2cm} Y_n=X_1+X_2,+.....+X_n\] \[\text {Clearly}\hspace {0.5cm} Y_1<Y_2<...........<Y_{n-1}<Y_n\] We want the joint pdf for \(Y_1,Y_2,.......,Y_n\). \begin {align*} f_{(X_1,X_2,....,X_n)}(x_1,x_2,....,x_n) &=\prod ^n_{i=1}f_X(x_i) =\prod ^n_{i=1}\lambda e^{-\lambda x_i}\\\\ &=\lambda ^ne^{-\lambda \sum \limits ^n_{i=1}x_i}\\ \end {align*}

\[X_1=Y_1,\hspace {0.2cm} X_2=Y_2-Y_1,\hspace {0.2cm} X_3=Y_3-Y_2,\hspace {0.2cm}..........,\] \[.........,\hspace {0.2cm} X_{n-1}=Y_{n-1}-Y_{n-2},\hspace {0.2cm} X_n=Y_{n}-Y_{n-1}\]

\[J= \begin {vmatrix} \frac {\partial X_1}{\partial Y_1} &..........& \frac {\partial X_1}{\partial X_n}\\\\ \frac {\partial X_2}{\partial Y_1} &..........& \frac {\partial X_2}{\partial Y_n}\\ .& &.\\ .& &.\\ .& &.\\\\ \frac {\partial X_n}{\partial Y_1}&..........& \frac {\partial X_n}{\partial Y_n}\\ \end {vmatrix} = \begin {vmatrix} 1&0&0& .......... &0&....&0\\ -1&1&0&.......... &0&...&0\\ 0&-1&1&0&........ &0&0\\ .& & & & &.&.\\ .& & & & &.&.\\ .& & & & &.&.\\ 0&0&0&0&......&-1&1\\ \end {vmatrix} \]
\[\implies \hspace {0.5cm} J=1\]
\[f_{Y_1,Y_2,\dots , Y_n}(y_1,y_2,......,y_n)=\lambda ^n e^{-\lambda Y_n},\hspace {0.5cm} Y_1<Y_2<.......<Y_n\]

\begin {align*} f_{Y_n}(y_n) &=\int \limits _{y_{n-1}}\int \limits _{y_{n-2}}........\int \limits _{y_2}\int \limits _{y_1}\lambda ^ne^{-\lambda y_n}dy_{y_1}dy_{2}.......dy_{n-1}\\\\ &=\lambda ^ne^{-\lambda y_n}\int \limits _{y_{n-1}}.........\int \limits _{y_3}\int \limits _{y_2}y_1\Big |^{y_2}_0\,dy_{2}\,dy_3.........\,dy_{n-1}\\\\ &=\lambda ^ne^{-\lambda y_n}\int \limits _{y_{n-1}}\int ^{y_{n-1}}_0\frac {y^{n-3}_{n-2}}{2\times 3\times 4\times ...(n-3)}\,dy_{n-3}\,dy_{n-1}\\\\ &=\lambda ^ne^{-\lambda y_n}\int \limits _{y_{n-1}}\frac {y^{n-2}_{n-1}}{2\times 3\times 4\times ....\times (n-3)(n-2)}\,dy_{n-1}\\\\ &=\lambda ^ne^{-\lambda y_n}\frac {y^{n-1}_{n-1}\Big |^{y_n}_0}{2\times 3\times 4\times .....(n-3)(n-2)(n-1)}\\\\ &=\frac {\lambda ^n e^{-\lambda y_n}y^{n-1}_n}{2\times 3\times 4\times ......\times (n-3)(n-2)(n-1)}\\\\ &=\frac {\lambda ^n\hspace {0.1cm} y^{n-1}_n\hspace {0.1cm}e^{-\lambda y_n}}{\Gamma (n)}\thicksim g(n,\lambda )\\\\ \end {align*}

\[X_i\thicksim g(1,\lambda )\]
\[Y_n=X_1+X_2+..........+X_n\thicksim g(n,\lambda )\]

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.