sample space & sigma field
Cond. Expect. of X given F :
The best reasonable
prediction of X
F = (Y) : E[X|Y]
E[X|Y](w) = E[X|Y (Y(w)]
E[X|B] = 1/P(B) BXdP
- Basic Stochastic Processes, Brzezniak and Zastawniak
properties of distributions
of future increments
w.r.t. the past
E[ future | p
(2), (2) 1
1.[7pts] Let and be the lines with parametric equations
Find the distance between and .
2.[7pts] If and are (distinct) position vectors, show that the vector
is a normal vector to the plane through the points and .
3.[10pts] Show that
1. Let be a point not on the plane that
passes through the points , and .
Show that the distance from the point to
the plane is
, and .
2. Use traces to sketch the quadric surface .
3. Let be a curve and be a point of curve.
(a) Find the uni
2010 (II) 2
1. Suppose that the temperature at a point in space is given by .
Find the rate of change of temperature at the point in the direction toward the point .
(a) In which direction does the temperature increase fastest at ?
(b) Find the maximum r
1. Evaluate the integral
where is the enclosed by the ellipsoid
2. Find the tangent plane to the surface with vector equation
sin cos , at the point .
3.(a) Evaluate the double integral
where , and is the disk .
(b) Evaluate the flux of the
4. Let F cos i cos j sin k . Find a function such that F and evaluate the line
F r, where
is the curve given by r isin j k .
5. Let .
(a) Show that the vector field
(b) Find a function such that .
(b) [4pts] Evaluate
the curve shown in the figure.
and for which the line
(a) [6pts] Find the curve between the points
reaches its minimum
value if it exists. If not, explain why.
2009 II 3
(a) Determine whether or not the vector field is conservative. If it is conservative, find a function
such that .
2. (a) Let be the region bounded by a positively oriented, piecewise-smooth, simple closed curve
2009 (II) 1
1.(a) Show that for any 3-dimensional non-zero vectors and , is orthogonal to .
(b) Show that , where and are non-coplanar vectors, and ,
. (Hint: , )
2. Find the volume of the tetrahedron with sides , , and .
3. Find parametri
2009 (II) 2
1. Let be a surface with equation and be a point on . Suppose is differentiable
State and prove the equation of the tangent plane to the surface at a point .
2. Find parametric equations for the tangent line to the curve of intersection
1. Assume that and have continuous second-order partial derivatives. Show that
any function of the form is a solution of the wave equation
2. Suppose that over a certain region of space the electronical potential is given
(a) Find th