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)
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)
