August 26th, 2010
Some time ago, in a half of the july (Bachelor exam -> exam for the second degree studies -> peace of mind) I realized that I commited myself to write article about Power Series (and things related). And now it is done, it is available under link below:
Power Series - convergence, Taylor's theorem (for this moment it is only available in Polish).
It contains a 7 examples of solving "Power Series" problems with detailed descriptions (Cauchy-Hadamard theorem, radius of convergence, surface of convergence, Taylor's theorem, differentiation and integration of power series, computing the sum of the power series. If I get motivated myself I will try to translate whole article to English.
Categories:
Mathematics. Tags:
examples,
mathematical analisys,
series theory.
May 3rd, 2010
In polish it's called "Prawo leniwego statystyka" ("Law of the Unconscious Statistician") and it is a property of expected value of discrete random variable. It says:
If 
is an random variable with values 
and its probabilities =P(X=x_i))
for each 
then for any function 
:
)=\sum_{i}g(x_i)f(x_i)=\sum_{i}g(x_i)P(X=x_i))
Proof:
f(x_i)=\sum_{j}\sum_{i:g(x_i)=y_j}g(x_i)f(x_i)=)
=y_j}f(x_i)=\sum_{j}y_jP(g(X)=y_j)=E(g(X))=Eg(x))
.
Example:
Find the expected value of variable ^X)
, where has following distribution:
=0,3&space;\quad&space;P(X=1)=0,1&space;\quad&space;P(X=3)=0,6)
Solution:
^X]=(-1)^0\cdot&space;0,3&space;+&space;(-1)^1\cdot&space;0,1+(-1)^3\cdot&space;0,6=)
.
Categories:
Mathematics. Tags:
examples,
probability,
theorems.
April 5th, 2010
There's new theorem & its proof (in polish) in Computer Science section which applies to the theory of automata. It's about equivalence of DFA and NFA, known as Scott Theorem (for me).
Categories:
Computer Science. Tags:
automata theory,
theorems.
