Download Descriptive Indices - National University of Singapore

Facts about Binomial & Poisson distribution
Fact 1 : If X ~ Bin (n, p), then
E ( X )  np
var( X )  np(1  p)
Fact 2 : If X 1 ~ Bin (n1 , p),
X 2 ~ Bin (n2 , p) independen tly,
then X 1  X 2 ~ Bin (n1  n2 , p)
Fact 3 : If X ~ Poisson ( ), then
E ( X )  var( X )  
Normal Approximation to Binomial
Distribution
p= 0.4, n=5
0.3
p= 0.4, n=10
p= 0.4, n=25
0.2
0.1
0.2
0.1
0.1
0 1 2 3 4 5
Bin (n, p)  N np, np(1  p) if n is large.
Demo: http://www.ruf.rice.edu/~lane/stat_sim/index.html
Normal Approximation to Poisson Distribution
Poisson ( )  N ,   if n is large.
Topic 8: Normal Distribution
A continuous random variable is one that can
take on any value within an interval.
The distribution of a continuous variable X is given
by a probability density function f (x) satisfying
(i) f ( x)  0 for all x  (,)
b
(ii) P(a  X  b)   f ( x)dx
a
(iii)



f ( x)dx  1.
Note that it is the integral of f ( x), i.e.,
the area under the density curve, and
not f ( x) itself, that gives us the probabilit y
In particular , P( X  x)  0  f ( x) for all x
Can think of the probability density f(x) as the
relative frequency histogram of a very large
sample
Expectation defined similarly as in discrete
but with integral instead of summation:

E( X ) 
 x f ( x)dx

Can again interpret expected value E(X) as the
long-run average of X under repeated sampling
The most well known continuous distribution
is the normal distribution.
Standard normal density
1
 z2
2
1
  z  
 ( z) 
e ,
2
 Bell-shaped
 Symmetric about 0
 Mean = 0, variance = SD = 1
 Well tabulated
From N (0,1) to N (  ,  2 ) and vice versa
 If Z ~ N (0,1) , then X    Z ~ N (  ,  2 )
2
 Conversely, if X ~ N (  ,  ) , then
X 
Z 
~ N (0,1) (Standardization)

Normally distributed random variables should be
standardized before looking up table
X  Blood Pressure ~ N (  129 mm Hg ,   19.8 )
2
2
X  129 150  129 

P( X  150)  P

  P( Z  1.06)  0.145
19.8 
 19.8
Demo: http://www.isds.duke.edu/sites/java.html
The standard normal density curve
If X ~ N (  ,  )
2


X 
P( | X   |  2 )  P Z 
 2 





within 2 SD from the mean
 0.954
P( | X   |  3 )  0.997


within 3 SD from the mean
This is the empirical rule
Normal distribution is often used to model
continuous measurement data such as
weight, height, blood pressure, etc.
The use of normal distribution is often
justified by the Central Limit Theorem
which says that the sum/average of a large
number of independent and identically
distributed variables is approximately
normally distributed.
Measurement error = sum of many indep
smaller errors
IQ : determined by many genetic &
environmental factors
Demo: http://www.ruf.rice.edu/~lane/stat_sim/index.html