文档介绍：Outline
Expectation of Random Variable
Moments of random variable and random vector
Chapter 3: Numerical Characteristics of
Distributions of Random Variables
Jiangsheng Yu
School of Electronics Engineering puter Science
Peking University, Beijing 100871, China
Probability Theory and Mathematical Statistics, 2007-2008
Jiangsheng Yu Numerical Characteristics of Distributions
Outline
Expectation of Random Variable
Moments of random variable and random vector
Outline of topics
1 Expectation of Random Variable
Definition of expectation
Examples of expectation
Some properties of expectation
2 Moments of random variable and random vector
Moments of random variable
Variance and its properties
Markov inequality, Chebyshev inequality and Lyapunov inequality
Moments of random vector
Coefficient of correlation and two types of regression
Double expectation
Coefficient of correlation and its properties
Two types of regression and their properties
Jiangsheng Yu Numerical Characteristics of Distributions
Discrete case: let P(X = xi) = pi be the probability of X,
X
E[g(X)] = pig(xi) (1)
i
P
whenever pi|g(xi)| < ∞.
i
Continuous case: let f(x) be the density of X,
Z +∞
E[g(X)] = f(x)g(x)dx (2)
−∞
R +∞
| | ∞
whenever −∞ f(x) g(x) dx < .
Outline Definition of expectation
Expectation of Random Variable Examples of expectation
Moments of random variable and random vector Some properties of expectation
Expectation: extension of mean
X is a random variable (rv) and g(X) is a single-valued function,
the expectation of g(X) is defined as
Jiangsheng Yu Numerical Characteristics of Distributions
Continuous case: let f(x) be the density of X,
Z +∞
E[g(X)] = f(x)g(x)dx (2)
−∞
R +∞
| | ∞
whenever −∞ f(x) g(x) dx < .
Outline Definition of expectation
Expectation of Random Variable Examples of expectation
Moments of random variable and random vector Some properties of expectation
Expectation: ex