Page 223 - DMTH404_STATISTICS

Unit 15: Exponential Distribution and Normal Distribution

Solution. Notes

Let X denote the quantity of petrol taken by a vehicle. It is given that X ~ N(20, 10).

-
æ 25 20ö
P
 (X  25 ) = P z  ÷ = P (z  0.5 )
ç
è 10 ø
- Pb £ £ 05g
=
.
-
.
= 05000. 0 z . = 05000 01915 03085
.
Let N be the total number of vehicles taking petrol on that day.
 0.3085×´ N = 100 or N = 100/0.3085 = 324 (approx.)
15.3.6 Normal Approximation to Binomial Distribution
Normal distribution can be used as an approximation to binomial distribution when n is large
and neither p nor q is very small. If X denotes the number of successes with probability p of a
success in each of the n trials, then X will be distributed approximately normally with mean np
and standard deviation npq .

X - np
Further, z = ~ N (0,1 ).
npq
It may be noted here that as X varies from 0 to n, the standard normal variate z would vary from
- ¥ to ¥ because

lim æ - np ö æ np ö
when X = 0, ç ÷ = lim - ÷ = - ¥
ç
n ® ¥ è npq ø n®¥ è q ø

lim æ n np ö lim æ nq ö lim æ nq ö
-
and when X = n, ç ÷ = ç ÷ = ç ÷ = ¥
n ® ¥ è npq ø n ® ¥ è npq ø n ® ¥ è p ø

Correction for Continuity

Since the number of successes is a discrete variable, to use normal approximation, we have make
corrections for continuity. For example,

æ 1 1ö
P(X £ X £ X ) is to be corrected as P X - £ X £ X + ÷ , while using normal approximation
ç
1 2 è 1 2 2 2 ø
to binomial since the gap between successive values of a binomial variate is unity. Similarly,
æ 1 1ö
P(X < X < X ) is to be corrected as P X + £ X £ X - ÷ , since X < X does not include X and
ç
1 2 è 1 2 2 2 ø 1 1
X < X does not include X .
2 2

Note The normal approximation to binomial probability mass function is good
when n  50 and neither p nor q is less than 0.1.

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