1


2


3

 Note: The joint density f(x,y) = C^{2} e^{1/2 ( x}^{2}^{
+ y}^{2}^{)} is rotationally invariant – the height
depends only on the radial distance from (0,0) and not on the angle.

4


5


6


7


8


9


10


11

 If Z = aX + bY, where a and b are arbitrary, we can define a new
variable:
 So Z’» N(0,1) and Z » N(0, Ö a^{2} + b^{2 }^{2}).

12


13


14


15

 Claim:
 Consider an experiment that is
 repeated independently n times
 where the i^{th}
outcomes has the probability p_{i}
for 1 · i · m.
 Let N_{i }=_{ }#
of outcomes of the i^{th}^{ }type (N_{1}+…+N_{m}
= n).
 Then for large n:

16

 Note: The claim allows to “test” to what extent an outcome is consistent
with an a priory guess about the actual probabilities.

17


18

