Objectives:
To study the nature of normal probability distribution and probability curve, to study the properties of normal distribution, central limit theorem and its applications.
Introduction
Normal distribution or Gaussian probability distribution is one of the most commonly used distribution. It is important due to wide use. The variables such as intelligence quotient, height of person, weight of person, errors in measurement of physical quantities follow normal distribution. Normal distribution is useful in statistical quality control, statistical inference, reliability theory, operations research, educational and psychological statistics. In the theory of sampling, designs of experiment also, normal distribution is very much useful. Normal distribution is viewed as a limiting distribution of several distributions such as Binomial, Poisson.
Definition of Normal Distribution
A continuous random variable X is said to follow normal distribution with parameters μ and o2, if its p.d.f. is,
I
z(1 − x).
1
f(x) =
e
0 < D'∞ > > co- '∞o > X > ∞!
σ2π
= 0
; otherwise.
Note:
1. A r.v. X follows normal distribution with parameters μ and o2 is symbolically written as X N (u, o2). Support of distribution is (-∞, ∞). Parameter space for μ is (-∞, ∞) and for σ is (0, ∞).
2. If μ = 0, σ2 = 1, then the normal variable is called as standard normal variable (S.N.V.) i.e. N (0, 1). Generally, it is denoted by Z. The p.d.f. of Z is,
1
f(z) =
√27
e-z2/2 ; - ∞ <Z<∞
with 3.141159 and e = 2.71828
3. The probability density curve of N (H, 2) is bell-shaped, symmetric about μ and mesokurtic as shown in the following figure. Hence, probability density curve of N (0, 1) is symmetric about zero.
01 02
Fig 4.1 (a)
02
å±± H2
H1<H2
0102
Fig. 4.1 (b)
4. We verify that the f(x) is p.d.f. as follows.
(i) Note that
f(x) =
202
01
(ii)
0
|f(x) dx = [] √2Ï€
σπ
( - x) :
-1
> 0 VX
e 202 (X-H)2
dx, put y=x=μ
f(x) dx =
σ dy =
e
dy
b/2
Since
Since e-ax2 xb-1 dx =
2ab/2
1/2
√2Ï€ 2√1/2
= 1
√2Ï€
(:: 1/2 = √F)
5. μ is called as location parameter and o is called as scale parameter.
1
0.399
6. The max. height of probability density curve =
E
σ2π
σ
Graph of normal distribution for N(0, 1) and N(2,5)
U.1
3:0
mu=0.var=1
nu=2,v=5
2
0
2
1
6
8
Fig. 4.2: Graph of Normal Distribution for N(0, 1) and N(2, 5)
0.399
-1
The standard normal
curve N (0, 1)
Fig. 4.3 (a)