Lecture Note STAT5126
Jiancheng Jiang
Chapter 1. Probability Theory
Probability theory: the foundation upon which all of statistics
is built
providing a means for modeling populations,
experiments, or almost anything else that could be
considered a rand
Lecture Note STAT5126
Jiancheng Jiang
1.3 Conditional Probability and Independence
Previously a sample space was defined and all probabilities were
calculated with respect to that sample space.
In many instances, we are in a position to update the sampl
Lecture Note STAT5126
Jiancheng Jiang
Chapter 2 Transformations and Expectations
We are often concerned with the behavior of functions of a
random variable X.
2.1 Distributions of Functions of a Random Variable
If X is a random variable with cdf FX(x),
Lecture Note STAT5126
Jiancheng Jiang
Chapter 3 Common Families of Distributions
3.1 Introduction
Statistical distributions are used to model populations; as
such, we usually deal with a family of distributions rather
than a single distribution.
This fa
Lecture Note STAT5126
Jiancheng Jiang
2.3 Moments and Moment Generating Functions
Definition 2.3.1 For each integer n, the nth moment of X (or
(), , is = . The nth central moment of X, , is
= ( ) , where = 1 = .
Definition 2.3.2 The variance of a rand
Lecture Note STAT5126
Jiancheng Jiang
3.3 Continuous Distributions
Uniform Distribution
The continuous uniform distribution is defined by
spreading mass uniformly over an interval [a, b]. Its pdf is
given by
Moments:
Gamma Distribution
The gamma func
Age
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
28.2
1.502794
27
22
8.231144
67.75172
6.283182
2.366899
37
21
58
846
30
Stem-and-Leaf Display
Stem unit: 1
Statistics
Sample Size
30
M