NORMAL APPROXIMATION OF BINOMIAL

Подпись: (6.3.1) image209

Here we shall consider in detail the normal approximation of a binomial variable as an application of the Lindeberg-Levy CLT (Theorem 6.2.2). In Definition 5.1.1 we defined a binomial variable X as a sum of i. i.d. Ber­noulli variables (T,): that is, X = Z”=1F,-, where Yt = 1 with probability p and = 0 with probability q = 1 — p. Since (F,) satisfy the conditions of the Lindeberg-Levy CLT, with EYt = p and VY, = pq, we can conclude

image210

FIGURE 6.1 Normal approximation of B(b,0.5)

As we stated in the last paragraph of Section 6.2, we may replace A

A A

above by ~. Or we may state alternatively that X/n ~ N(p, pq/n) or that

д

X ~ (np, npq). We shall consider three examples of a normal approxima­tion of a binomial.

EXAMPLE 6.3.1 Let X be as defined in Example 5.1.1. Since EX = 2.5 and VX = 1.25 in this case, we shall approximate binomial X by normal X* ~ N{2.5, 1.25). The density function f(x) of iV(2.5, 1.25) is, after some rounding off,

(6.3.2) f(x) = g-g exp [-(ж – 2.5)2/2.5].

Using (5.1.12) and (6.3.2), we draw the probability step function of bi­nomial X and the density function of normal X* in Figure 6.1. The figure suggests that P(X = 1) should be approximated by P(0.5 < X* < 1.5), P(X = 2) by P(1.5 < X* < 2.5), and so on. As for P(X = 0), it may be approximated either by P(X* < 0.5) or P(—0.5 < X* < 0.5). The same is true of P(X = 5). The former seems preferable, however, because it makes the sum of the approximate probabilities equal to unity. The true probabilities and their approximations are given in Table 6.1.

EXAMPLE 6.3.2 Change the above example to p = 0.2. Then EX = 1 and VX = 0.8. The results are summarized in Table 6.2 and Figure 6.2.

example 6.3.3 If 5% of the labor force is unemployed, what is the probability that one finds three or more unemployed workers among

table 6.1 Normal approximation of В(5, 0.5)

X

Probability

Approximation

0 or 5

0.03125

0.0367

1 or 4

0.15625

0.1500

2 or 3

0.31250

0.3133

table 6.2 Normal approximation of B(5, 0.2)

k

P(X = k)

Approximation

0

0.3277

0.2877

1

0.4096

0.4246

2

0.2048

0.2412

3

0.0512

0.0439

4

0.0064]

0.0026

5

0.0003J

image211

FIGURE 6.2 Normal approximation of B(5,0.2)

twelve randomly chosen workers? What if 50% of the labor force is unem­ployed?

Let X be the number of unemployed workers among twelve workers. Then X ~ B(12, p), where we first assume p = 0.05. We first calculate the exact probability:

(6.3.3)

Подпись: = 1 Подпись: 19^ 20 image214

P(X > 3) = 1 – P(X = 0) – P(X = 1) – P(X = 2)

image215

= 0.02.

P(Z > 2.52) = 0.0059,

where Z is N(0, 1). This is a poor approximation.

Подпись: (6.3.5) Подпись: P(X > 3) = 1 - image218 Подпись: 12 image220

Next, put p = 0.5. Then the exact probability is given by

and the approximation using X* ~ N(6, 3) yields

(6.3.6) P(X > 3) = P(X* > 2.5) = P(Z > -2.02) = 0.9783,

which is a good approximation.

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