Transcript 525201 Statistics and Numerical Method Part I: Statistics

525201
Statistics and Numerical Method
Part I: Statistics
Week III: Random Variables and
Probability Distribution
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สมศักดิ์ ศิวดำรงพงศ์
[email protected]
Random Variables
Controllable Variables
Output
Input
Uncontrollable Variables
Random Variable : A Numerical variable whose measured
value can change from one replicate of
the experiment to another
3-2 Random Variables
Discrete random variables
 Continuous random variables

3-3 Probability
The chance of “x”
 A degree of belief
 A relative frequency between “event
frequency” to the “outcome frequency”

Histogram of Compressive
Strength
Histogram of Compressive
Strength
0.275
22
0.2125
0.175
14
0.125
0.075
0.0250.0375
17
10
0.05
0.025
2
3
6
4
2
3-4 Continuous Random Variables

Cumulative Distribution Function (cdf)
x
F ( x )  P( X  x ) 
 f (u)du

for
  x  
Continuous Random Variables

Probability Density Function (pdf)
b
P(a  x  b)   f ( x)dx
a
when
1) f ( x)  0

2)  f ( x)dx  1

Continuous Random Variables

Mean and Variance
Example 3.5
3-5.1 Normal Distribution (Gaussian)
Normal Distribution
Normal Distribution
Normal Distribution
Normal Distribution
Normal Distribution
t-Distribution
When  is unknown
 Small sample size
 Degree of freedom (k) = n-1
 Significant level = 
 t, k

t-Distribution
3-7 Discrete Random Variables
• Probability Mass
Function (pmf)
Discrete Random Variables

Cumulative
Distribution
Function (cdf)
Discrete Random Variables

Mean and Variance
3-8 Binomial Distribution

A Bernoulli Trial
Binomial Distribution
Binomial Distribution
Example 3-28 Bit transmission errors: Binomial Mean and Variance
3-9 Poison Distribution
The random variable X that equals the number of
events in a Poison process is a Poison random variable
with parameter >0, and the probability mass function
of X is
 x
f ( x) 

e
x!
The mean and variance of X are
E ( x)   and V ( x)  
3-9 Poison Distribution
3-9 Poison Distribution
3-9 Poison Distribution
3-10 Normal Approximation to the
Binomial and Poisson Distributions

Normal Approximation
to the Binomial
3-10 Normal Approximation to the
Binomial and Poisson Distributions
3-10 Normal Approximation to the
Binomial and Poisson Distributions

Normal Approximation to the Poisson
3-13 Random Samples, Statistics
and the Central Limit Theorem
3-13 Random Samples, Statistics
and the Central Limit Theorem
x1  x2  ...  xn
X
n
E( X )  
V (X ) 

2
n
3-13 Random Samples, Statistics
and the Central Limit Theorem
3-13 Random Samples, Statistics
and the Central Limit Theorem
Q &A