Introduction to Probability and Statistics for Engineers and Scientists

6.3The Central Limit Theorem 209

Enter the probabilities and number of random

variables to be summed. The output gives the mass

function of the sum along with its mean and

variance.

0 100

i

0.030

0.025

0.020

0.015

0.010

0.005

0.000

p(i)

Central Limit Theorem

Start

Quit

.25

.15

.1

.2

.3

P0

P1

P2

P3

P4

n = 100

(f)

Mean = 215.

Variance = 252.75

200 300 400

FIGURE 6.2 (continued)

Because

E[Xi]=p, Var(Xi)=p(1−p)

it follows from the central limit theorem that fornlarge

X−np

√

np(1−p)

will approximately be a standard normal random variable [see Figure 6.3, which graphically
illustrates how the probability mass function of a binomial (n,p) random variable becomes
more and more “normal” asnbecomes larger and larger].

EXAMPLE 6.3c The ideal size of a first-year class at a particular college is 150 students.
The college, knowing from past experience that, on the average, only 30 percent of those
accepted for admission will actually attend, uses a policy of approving the applications of
450 students. Compute the probability that more than 150 first-year students attend this
college.

SOLUTION LetX denote the number of students that attend; then assuming that each
accepted applicant will independently attend, it follows thatXis a binomial random