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Solutions to midterm problems related to finding the jacobian and checking the independence of random variables in probability distributions. Topics include the jacobian matrix, marginalization, and the characteristic function of a poisson distribution.
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Midterm Solutions EE Fall
Problem
X Y unif
Z X Y Let U X and V X Y and form the jacobian for fU V u v
fU V
jU j
fX Y u v u
Marginalize with resp ect to u and employ the indep endence of X and Y
fV v fZ z
jU j
fX ufY z udu
fZ z
jU j
fY z udu
The domain for fY is b ounded away from by the z u term First consider z for fY z u juj z Therefore the integral ab ove can b e completed for z by splitting it up into when z u and u z so for z
fZ z
z
du
du
fZ z
z
du
fZ z
l nz
now similarly for z
fZ z
du
z
du
fZ z
z
du
fZ z
l nz so
fZ z
l njz j
it is unde ned for z but that s ok b ecause it happ ens with a probability of The integral can b e evaluated in a closed form though b since X is indep endent of Y
E X Y E X E Y
Problem
P T t
t
F t
t t
fT t
dFT dt
t b
P atl eastbul bw or k ing att g iv enw or k ing att
P w or k ing att j w or k ing att
P att att P att
P t t
t t
t
P t
P w or k ing att jal l w or k ing att
P atl eastbul b
c
Z minT T T T
P Z t
P T tP T tP T tP T t
fr r r fX Y r cos r sin
Problem extra credit
The characteristic function of a Poission rv is
Gs exps
The sum of two indep endent rv s is the pro duct of their characteristic func tions H s expas expbs expa bs so converting back the characteristic function which is in standard form
fZ k a bk k
expa bk