1


2


3


4


5


6


7

 Claim: a continuous 11 function has to be strictly monotonic, either
increasing or decreasing
 Pf: If a continuous function g(x) is not strictly monotonic then there
exist x_{1 }< x_{2 }< x_{3} such that g(x_{1})
· g(x_{2 }) ¸ g(x_{3 }) or g(x_{1 }) ¸ g(x_{2
}) · g(x_{3 }).
 This implies by the meanvalue theorem that the function cannot be 11.

8


9


10


11


12


13


14


15


16

 Problem: Suppose a point is picked uniformly at random from the
perimeter of a unit circle.
 Find the density of X, the xcoordinate of the point.

17


18

 Notice, that is not necessary to find the density of Y = g(X) in order
to find E(Y).

19

 Problem: Suppose a point is picked uniformly at random from the surface of a unit
sphere. Let Q be the latitude of this point as seen on the diagram. Find
f_{Q}(q).
