• Under certain conditions, the normal

distribution can be used to approximate the

binomial distribution and the Poisson

distribution.

Figure 4-13 Normal approximation

to the binomial distribution.

Normal

Approximation

to the Binomial

Distribution

If X is a binomial random variable with parameters n and p,

is approximately a standard normal random variable. To

approximate a binomial probability with a normal

distribution, a continuity correction is applied as follows:

and

1

X npZ

np p

0.50.5

1

x npP X x P X x P Z

np p

0.50.5

1

x npP x X P x X P Z

np p

Note: The approximation is good and 5np 1 5.n p

Example 1

In a digital communication channel, assume that the number of bits

received in error can be modeled by a Binomial random variable, and

assume that the probability that a bit is received in error is If

16 million bits are transmitted, what is the probability that 150 or

fewer errors occur?

51 10 .

Solution

Let the random variable X denote the number of errors. Then X is a

binomial random variable and since

and is much larger, the

approximation is applied.

150 150.5P X P X

5 5

160 150.5 160

160 1 10 160 1 10

0.75 0.227

XP

P Z

6 516 10 1 10 160 5np 1n p

Example 2

Again consider the transmission of bits in Example 1. To judge how well

the normal approximation works, assume only n = 50 bits are to be

transmitted and that the probability of and error is p = 0.1. The exact

probability that 2 or less errors occur is

Based on the normal approximation,

4850 49 2

50 50 502 0.9 0.1 0.9 0.1 0.9 0.112

0 1 2P X

5 2.5 52

50 0.1 0.9 50 0.1 0.9

XP X P

1.18 0.119P Z

Example 3

The manufacturing of semiconductor chips produces 2% defective

ships. Assume the chips are independent and that a lot contains 1000

chips.

(a) Approximate the probability that more than 25 chips are defective?

(b) Approximate the probability that between 20 and 30 chips are

defective?

Solution

Let X denote the number of defective chips

(a)

1000 0.02 20np

1 1000 0.02 0.98 4.43np p

25 1 25P X P X

25 0.5 201

4.43

1 1.24 1 0.89251 0.10749

P Z

P Z

Solution

(b) 20 30 21 29P X P X

21 0.5 20 29 0.5 20

4.43 4.43

0.11 2.14

2.14 0.11

0.98382 0.54379

0.44003

P Z

P Z

P Z P Z

Normal Approximation to the Poisson Distribution

If X is a Poisson random variable with and

is approximately a standard normal random variable. The same

continuity correction used for the binomial distribution can also be

applied. The approximation is good for

XZ

5

E X V X

Example 4

Assume that the number of asbestos particles in a squared meter of

dust on a surface follows a Poisson distribution with a mean of 1000. If

a squared meter of dust is analyzed, what is the probability that 950 or

fewer particles are found?

Solution

The probability can be approximated as

950.5 1000

950 950.51000

P X P X P Z

1.57 0.058P Z

Example 5

A high-volume printer produces minor print-quality errors on a test

pattern of 1000 pages of text according to a Poisson distribution with a

mean of 0.4 per page.

(a) What is the mean number of pages with errors (one or more)?

(b) Approximate the probability that more 350 pages contain errors

(one or more).

Solution

Let X denote the number of minor errors on a test pattern of 1000

pages of text

(a)

The mean number of pages with one or more errors is

~ Poisson 0.4X

0.4 00.4

0 0.6700!

eP X

1 1 0 1 0.670 0.330P X P X

1000 0.330 330

Solution

(b) Let Y denote the number of pages with errors

~ Bin 1000, 0.330Y n p

350 1 350P Y P Y

350 0.5 3301

1000 0.330 0.670

1 1.38

1 0.9162

0.0838

P Z

P Z