Summary FEB21005 Probability Theory

FEB210055 Probability Theory


LECTURE 1
·
Recap Introduction to Statistics

Axioms of probability
① p(A) = 0 for any event A

② p(s) 1 S = :
sample space

③ For any countable collection of mutually exclusive events :
D(Ai) = Plai)
↳
p(A)) = 1 -



p(A)
Conditional probability

PlAIB) =
PIAB if P(B) > o




In dependent Events
A and B independent if : p(A) p(B)
p(A 1 B) = .




probability density function (pdf)
·




Discrete pdf
fx(x) p(x x)
= =




f(x) for all xER
[ ax
10


f(xi) = 1
,


Continuous pdf
fx(x) =
Fx(x)

( f(x) for all xER
:0
A


Sf(x)dx =
1
- A

·
Cumulative density function (CDF)
#x(x) =
p(x = x)) -
right-continuous)
↳ F(-0) =im F(x) = starts at


↳:
o


F( + 0) =
m F(x) = 1 ends at 1

F(X) is non-decreasing (if x, < x2 then F(x) =
F(x2))
·
Relation continuous pdf and CDF
p(x = x) =
Ex(x) = ()ds and placXb) =Sels) as

, LECTURE 2

·
Expected Value

E(X) &xi f(xi)
=

A
·




if X is discrete

E(x) (x . f(x)dx
=
if X is continuous
-
A



If u(X) is a random variable :

E(u(x)) =
Eu(xi) ·




f(xi) if X is discrete

E(u(x)) = u(x) -



f(x)dx if X is continuous

properties of the expected value

E(c) =
c



E(aX + b) =
af(x) + b

E(a g(x) .
+ b .


n(x)) =
a
-


E(g(x)) + b .
E(h(x)
o
Variance

Var(x) =
E((X -



E(X())
02 : variance and 0 : standard deviation

properties of the variance

Var(x) = 0


Var(aX) =
a2Var(x)
Var(X + b) =
Var(x)
Var(aX + b) = a Var(x)

Var(x) =
E(xz) -


(E(X))2
·
Moments

E(X) is the th moment mi of RV X
↳ moment : mi E(x)
↳ first
=




second moment Mi :
=
E(X)
Central moments
E (X-M(4) ,
where M
=
Mi ,
is the th central moment Mr of RV X

↳ central moment : M , E(X m) E(x)
↳ first
= -
= -


M = 0


second central moment E((X M) )
2
:
Mz =
-
=
Var(x)