chapter 4.7.pdf
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• 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