Normal Distribution

1. 1.13 Normal distribution

2. Normal distribution
A continuous random variable X has a normal distribution and it is
referred to as a normal random variable if its probability density is
given by
1  ( x   ) / 2
2 2
f ( x ;  , )  for    x  
2
e
2 
w h ere -  <  <  an d  > 0 are th e p aram eters
of th e n orm al d istrib u tion .
The normal distribution is also known as Gaussian distribution
because the Gaussian function is defined as
2 2
 ( xb ) / c
f ( x)  a e

3. Normal distribution
Normal probability density function for selected
values of the parameter  and 2

4. Mean of a Normal distribution
 
1 2
u / 2
   ue du    f ( x ; ,  ) dx .
2
2  
 

5. Variance of a Normal distribution

2  
2   
2  
2
2
u / 2
 ue du  
2 2
u / 2 u / 2
  e e du
2  
 
 0

 2 0
 0 

2

6. Moment Generating Function
E(X )  
V ar ( X )  
2
 t
2 2
 t
M X
(t )  e 2

7. Standard Normal Distribution
A normal random variable with mean 0 and variance 1 is called
standard normal random variable.
If Z be a standard normal random variable, then the density
function of Z is given by
1 2
z /2
f (z)  e ,   z
2

8. Standardizing a random variable
A random variable can be made standard by
standardizing it. That is, if X is any random
variable then X  E ( X ) is a standard random
S .D .
variable.

9. The distribution function of a standard
normal variable is
z
1 2
t / 2
F (z) 
2
e dt  P ( Z  z )

0 z a b
The standard normal probabilities The standard normal probability
F(z) = P(Z  z) F(b) - F(a) = P(a  Z  b)

10. If a random variable X has a normal distribution
with the mean  and the standard deviation ,
then, the probability that the random variable X
will take on a value less than or equal to a, is
given by
 X  a  a a
P( X  a)  P    P Z    F 
         
Similarly
b  a 
P (a  X  b)  F   F 
     

11. The Normal Distribution
Although the normal distribution applies to continuous
random variable, it is often used to approximate
distributions of discrete random variables.
For that we must use the continuity correction according to
which each integer k be represented by the interval from k
– ½ to k + ½.
For instance, 3 is represented by the interval from 2.5 to
3.5, “at least 7” is represented by the interval from 6.5 to 
and “at most 5” is represented by the interval from - to
5.5. Similarly “less than 5” is represented by the interval
from - to 4.5 and “greater than 7” by the interval from
7.5 to .

12. The normal Approximation to the
Binomial Distribution
• The normal distribution can be used to approximate the
binomial distribution when n is large and p the probability of a
success, is close to 0.50 and hence not small enough to use
the Poisson approximation.
Normal approximation to binomial distribution
Theorem If X is a random variable having the binomial
distribution with the parameters n and p, and if
X  np
Z 
np (1  p )
then the limiting form of the distribution function of this
standardized random variable as n   is given by

13. The normal Approximation to the
Binomial Distribution
z 1 2
t / 2
F (z)  

2
e dt    z  .
Although X takes on only the values 0, 1, 2, … , n, in the limit
as n   the distribution of the corresponding standardized
random variable is continuous and the corresponding probability
density is the standard normal density.

14. The normal Approximation to the
Binomial Distribution
A good rule of thumb for the normal approximation
For most practical purposes the approximation is acceptable for
values of n & p such that
Either p ≤ 0.5 & np > 5
Or p > 0.5 & n (1-p) > 5.