pfsl1.tex
Lecture 1. 9.10.2013
I. AXIOMATIC PROBABILITY THEORY
1. Length, area and volume. We shall mainly deal with area, as
this is two-dimensional. We can draw pictures in two dimensions, and our
senses respond to this; paper, whiteboards and computer
pfsl2.tex
Lecture 2. 10.10.2013
Denition. A -eld (or -algebra) A is a class containing the whole set,
closed under complements, and closed under countable disjoint unions
(the here is from the German Summe = sum the old-fashioned notation
for a union is a
pfsl3.tex
Lecture 3. 10.10.2013 (half-hour:problems)
3. Distributions and distribution functions
The distribution function F (x) := P (X x) of X is a Lebesgue-Stieltjes
measure function; it determines the corresponding Lebesgue-Stieltjes measure by (denot
pfsl4.tex
Lecture 4. 16.10.2013
Higher dimensions; joint and marginal distributions
If X = (X1 , . . . , Xn ) is a random variable taking values in n-dimensional
space a random n-vector then its distribution function F is dened as
above, but coordinatewis
pfsl5.tex
Lecture 5. 17.10.2013
6. Fisher F -distribution, F (m, n). This is dened as that of the ratio
F :=
U/m
,
V /n
where U 2 (m), V 2 (n) and U, V are independent. This has density
m 2 mn 2 n
x 2 (m2)
f (x) =
.
1
1
B ( 2 m, 1 n) (mx + n) 2 (m+n)
2
1
pfsl6.tex
Lecture 6. 17.10.2013 (half-hour: Problems)
2
2
The next obvious example is the Normal: if X N (1 , 1 ), Y N (2 , 2 ),
2
2
and X , Y are independent, then X + Y N (1 + 2 , 1 + 2 ). This is indeed
true, and can be proved as above (try it as an ex
pfsl8.tex
Lecture 8. 24.10.2013 (half-hour: Problems)
We form the probability generating function (PGF)
X
P (s), or PX (s), := E [s ] =
n
s P (X = n ) =
n=0
pn sn .
n=0
This is a power series in s, and since
pn = 1, it converges for s = 1. So
the radius o
pfsl9.tex
Lecture 9. 30.10.2013
We quote (see e.g. [GS], 7.2): if 1 p r,
(a) Lr Lp [true in any nite measure space, but not in general];
(b) convergence in rth mean implies convergence in pth mean
[as expected: the higher the moment, the more restrictive
pfsl10.tex
Lecture 10. 31.10.2013
Proof. When we subtract from each Xk , we change the mean from to
0 and the second moment from 2 to the variance 2 . So by the moments
property of CFs, Xk has CF 1 1 2 t2 +o(t2 ) as t 0. So X1 +. . .+Xn n
2
has CF
1
E exp
pfsl11.tex
Lecture 11. 31.10.2013 (half-hour: Problems)
In the general case, we use the Probability Integral Transformation (PIT,
IS, I). Let U1 , . . . , Un . . . be iid uniforms, Un U (0, 1). Let Yn := g (Un ),
where g (t) := supcfw_x : F (x) < t. By PI
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Lecture 12. 6.11.2013
Take expectations: as Ey = , Eg (y ) g (). So
g (y ) g () g (y ) Eg (y ) g ()(y ).
Square both sides:
[g (y ) g ()]2 [g ()]2 (y )2 .
Take expectations: as Ey = and Eg (y ) g (), this says
var(g (y ) [g ()]2 var(y ).
Regres
pfsl13.tex
Lecture 13. 7.11.2013
Denition. Two distribution functions F , G have the same type if
G(x) = F (a + bx)
for some a, b. Then if Y := (X a)/b, and X F , then Y G.
Stable laws.
The possible limit laws obtainable from centred and scaled random wal