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240 Chapter 4 Special Distributions
Proof One of the ways to verify Theorem 4.3.1 is to show that the limit of the
2
2
X − np
moment-generating function for √np(1
as n → ∞ is et /2 and that et /2 is also
− p)
&∞
2
the value of −∞ et z · √12π e−z /2 dz. By Theorem 3.12.2, then, the limiting pdf of
X −np
Z = √np(1−
is the function f Z (z) = √12π e−z
p)
for the proof of a more general result.
2 /2
, −∞ < z < ∞. See Appendix 4.A.2
Comment We saw in Section 4.2 that Poisson’s limit is actually a special case of
−λ k
Poisson’s distribution, p X (k) = e k!λ , k = 0, 1, 2, . . . . Similarly, the DeMoivre-Laplace
limit is a pdf in its own right. Justifying that assertion, of course, requires proving
2
that f Z (z) = √12π e−z /2 integrates to 1 for −∞ < z < ∞.
Curiously, there is no algebraic or trigonometric substitution that can be used to
demonstrate that the area under f Z (z) is 1. However, by using polar coordinates,
we can verify a necessary and sufficient alternative—namely, that the square of
& ∞ 1 −z 2 /2
√
dz equals 1.
−∞ 2π e
To begin, note that
% ∞
% ∞
% ∞% ∞
1
1
1
1 2
2
2
2
e−x /2 d x · √
e−y /2 dy =
e− 2 (x +y ) d x d y
√
2π −∞ −∞
2π −∞
2π −∞
Let x = r cos θ and y = r sin θ , so d x d y = r dr dθ . Then
% ∞% ∞
% 2π % ∞
1
1
1 2
2
2
e− 2 (x +y ) d x d y =
e−r /2 r dr dθ
2π −∞ −∞
2π 0
0
% 2π
% ∞
1
2
r e−r /2 dr ·
dθ
=
2π 0
0
=1
Comment The function f Z (z) = √12π e−z
2 /2
is referred to as the standard normal (or
Gaussian) curve. By convention, any random variable whose probabilistic behavior
is described by a standard normal curve is denoted by Z (rather than X , Y , or W ).
2
Since M Z (t) = et /2 , it follows readily that E(Z ) = 0 and Var(Z ) = 1.
Finding Areas Under the Standard Normal Curve
In order to use Theorem 4.3.1, we need to be able to find the area under the graph of
f Z (z) above an arbitrary interval [a, b]. In practice, such values are obtained in one
of two ways—either by using a normal table, a copy of which appears at the back
of every statistics book, or by running a computer software package. Typically, both
approaches give the cdf, FZ (z) = P(Z ≤ z), associated with Z (and from the cdf we
can deduce the desired area).
Table 4.3.1 shows a portion of the normal table that appears in Appendix A.1.
Each row under the Z heading represents a number along the horizontal axis of
f Z (z) rounded off to the nearest tenth; Columns 0 through 9 allow that number to be
written to the hundredths place. Entries in the body of the table are areas under the
graph of f Z (z) to the left of the number indicated by the entry’s row and column. For
example, the number listed at the intersection of the “1.1” row and the “4” column
is 0.8729, which means that the area under f Z (z) from −∞ to 1.14 is 0.8729. That is,
% 1.14
1
2
√ e−z /2 dz = 0.8729 = P(−∞ < Z ≤ 1.14) = FZ (1.14)
2π
−∞