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Probability, statistics and options I course of the University of Oxford

Past exam of the probability, statistics and options I course of the University of Oxford

2008

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The inverse change of variables is $x=2u+v$, $y=v$, with absolute <Jacobian determinant> two. Positivity of $x,y$ gives
$$
A=\{(u,v):v>0,\ 2u+v>0\}.
$$
The <change of variables for probability densities> gives
$$
f_{U,V}(u,v)=\tfrac14e^{-(2u+2v)/2}\cdot2
=\tfrac12e^{-u-v}\quad((u,v)\in A).
$$
For fixed $u$, integrate over $v>\max(0,-2u)$:
$$
f_U(u)=\tfrac12e^{-u}\int_{\max(0,-2u)}^\infty e^{-v}\,dv
=\begin{cases}\tfrac12e^{-u},&u\geq0,\\
\tfrac12e^u,&u<0.\end{cases}
$$
Thus $\boxed{f_U(u)=\tfrac12e^{-|u|}}$, the standard <Laplace distribution>.

Solution

Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / M / 1 / Solution

Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / M / 1 / Solution

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