Image Analysis, Classification and Change Detection in Remote Sensing, 3rd Edition

Random variables 45

sample mean (see Section 2.2),

¯

Z =

1

m

m

X

i=1

Z

i

,

is normally distributed with mean µ and varianc e σ

2

/m (Exercise 5). Theorem

2.3, on the other hand, justifies approximating the distribution of the mean

¯

Z with a normal distributio n having mean µ and variance σ

2

/m for la rge m,

even when the Z

i

are not no rmally dis tributed.

As an illustration of the Central Limit Theorem, the IDL code in Listing 2.1

calculates 10,000 sums of m = 12 random numbers uniformly distributed on

the interval [0, 1] and plots their histogram to a PostScript file; see Figure 2.1.

Note the use of the IDL function TOTAL(A,2), which sums a two- dimens ional

IDL array ...