Normal Distribution
It is important because of Central Limit Theorem (CTL), the CTL said that Sum up a lot of i.i.d random variables the shape of the distribution will looks like Normal. Normal P.D.F

Now we want to find c
This integral has been proved that it cannot have close form solution. However, someone gives an idea that looks stupid but actually very brilliant by multiply two of them.

reminds the function of circle which we can replace them to polar coordinate
Thus
Mean

By symmetry if g(x) is odd function g-x=-g(x) then -abgxdx=0 Variance

Notation
CDF is standard Normal CDF
by symmetric

,CDF , , All the odd moment of standard normal are zero. However, even moment is not easy to calculate by integral (Symmetry)

Then we say
Most of Statistics books will write the pdf then explain the mean and variance but it is not intuitive.
Standardization

Find PDF of
CDF:
The PDF is derivative of the CDF (using chain rule)
PDF:

Later we’ll show if independent

68-95-99.7% Rule
Because you can’t actually calculate the , somebody create a rule of thumb
The properties of variance

If you shift the variance by c, the mean also shift by c. Thus, the variance doesn’t change.
Remember to square. It is easy to validate if you think c is negative and don’t square it. The corresponding variance becomes negative which contradict the definition of the variance. iff for some

Variance only can be zero if and only if the probability is a constant. in general [It is equal if X,Y are independent]
To get the idea about this we let Y = X which means the two r.v.s are extremely dependent. Then

...standard deviation = square root of variance = sqrt(846) = 29.086 4. If we have the following data
34, 38, 22, 21, 29, 37, 40, 41, 22, 20, 49, 47, 20, 31, 34, 66 Draw a stem and leaf. Discuss the shape of the distribution. Solution: 2 3 4 5 6 | | | | | 219200 48714 0197 6
This distribution is right skewed (positively skewed) because the “tail” extends to the right. 5. What type of relationship is shown by this scatter plot?
45 40 35 30 25 20 15 10 5 0 0 5...

...•H.P.Gautam
The purpose of this article is not to explain any more the usefulness of normaldistribution in decision-making process no matter whether in social sciences or in natural sciences. Nor is the purpose of making any discussions on the theory of how it can be derived. The only objective of writing this article is to acquaint the enthusiastic readers (specially students) with the simple procedure ( iterative procedure) for finding the numerical value of a...

...real numbers t with the following
properties:
(1)
(2)
(3)
(4)
W0 = 0.
With probability 1, the function t → Wt is continuous in t.
The process {Wt }t≥0 has stationary, independent increments.
The increment Wt+s − Ws has the N ORMAL(0, t) distribution.
A Wiener process with initial value W0 = x is gotten by adding x to a standard Wiener
process. As is customary in the land of Markov processes, the initial value x is indicated
(when appropriate) by putting a...

...skewed-right distribution with a mean of 10 minutes and a standard deviation of 8 minutes. Suppose 100 flights have been randomly sampled. Describe the sampling distribution of the mean waiting time between when the airplane taxis away from the terminal until the flight takes off for these 100 flights. a) Distribution is skewed-right with mean = 10 minutes and standard error = 0.8 minutes. b) Distribution is skewed-right with mean = 10...

...Business Statistics
Chapter 7
Sampling and Sampling Distributions
6-1
Learning Objectives
In this chapter, you learn:
The concept of the sampling distribution
To compute probabilities related to the sample
mean and the sample proportion
The importance of the Central Limit Theorem
To distinguish between different survey
sampling methods
To evaluate survey worthiness and survey errors
7-2
Reasons for Drawing a Sample
Selecting a sample is less...

...SAMPLING DISTRIBUTIONS
|6.1 POPULATION AND SAMPLING DISTRIBUTION |
|6.1.1 Population Distribution |
Suppose there are only five students in an advanced statistics class and the midterm scores of these five students are:...

...X-bar Definition
1 x xi n i 1
Probability and statistics - Karol Flisikowski
n
Sampling Distribution of x-bar
How does x-bar behave? To study the behavior,
imagine taking many random samples of size n, and computing an x-bar for each of the samples. Then we plot this set of x-bars with a histogram.
Probability and statistics - Karol Flisikowski
Sampling Distribution of x-bar
Probability and statistics - Karol Flisikowski...

...or variability of the data about the measurements of central tendency.
MEASUREMENTS OF CENTRAL TENDENCY The appropriateness of using the mean, median, or mode in data analysis is dependent upon the nature of the data set and its distribution (normal vs non-normal). The mean (denoted by x) is calculated by dividing the sum of the individual data points (where Σ equals “sum of”) by the number of observations (denoted by n). It is the arithmetic...