1. Central Limit Theorem: Sampling Distribution.
Jacek Wallusch _________________________________ Mathematical Statistics for International Business
Lecture 6:
Central Limit Theorem: Sampling Distribution.

2. a probability distribution of a statistic
Getting Started ____________________________________________________________________________________________
definitions
Statistic
any real or vector-valued function of the observed random variables, describing a characteristic of the random variable
Sampling Distribution
a probability distribution of a statistic
Sample
a sequence of independent identically distributed random variables with a common distribution function or density
Population
distance measured in s.d. units
a collection of all independent identically distributed random variables with a common distribution function or density
Mathematical Statistics: 6

3. normal density function is symmetric with respect to the mean value
Normal Distribution ____________________________________________________________________________________________
probability
Probability, mean value and standard deviation
normal density function is symmetric with respect to the mean value
distance between mean value and any point at the normal curve: distance measured in s
distance measured in s.d. units
Mathematical Statistics: 6
Conversion formula z

4. Standard Normal Distribution ____________________________________________________________________________________________
probability
Probability, mean value and standard deviation
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 6
Do not confuse z with the z-score

5. SND ____________________________________________________________________________________________
central limit theorem
Assumptions:
sequence of independent and identically distributed variables
central moments, mean value and variance, defined as
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 6
mean value and variance are both FINITE

6. SND ____________________________________________________________________________________________
central limit theorem
Assumptions:
define additionally
where
Theorem states that:
the distribution of UT converges to a standard normal distribution function as T approaches infinity
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 6
mean value of a large number of iid random variables is approximately normally distributed

7. Not everything lies within 2 standard deviation of the mean value
Normal Distribution ____________________________________________________________________________________________
beware of misuse
Not everything lies within
2 standard deviation of the mean value
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Do not follow blindly
Mathematical Statistics: 6
ask your doctor about the normal distribution

8. CLT in Practice ____________________________________________________________________________________________
probability
Calculate the probability
MS Excel formula: standard normal distribution
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
use the value of z, and subtract it from 1
Mathematical Statistics: 6
Do not confuse z with the z-score

9. CLT in Practice ____________________________________________________________________________________________
examples
Exercises:
1. Use the data for EUR and USD, calculate the probability of obtaining different aims;
2. Use the data on expected salaries, calculate the probability that a female student claims to earn more than an average male student;
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 6
Do not confuse z with the z-score

10. CLT in Practice ____________________________________________________________________________________________
examples
Helpful formulas:
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 6
Do not confuse z with the z-score