14-ma`ruza. Variance and Standard Deviation

Theorem A.19 (limit theorem of de Moivre–Laplace)²¹ If the probability of the occurrence of an event A in n independent trials is constant and equal to p, 0 < p < 1, then the probability P (X = x) that, in these trials, the event A occurs exactly x times satisfies as n →∞ the relation

√

2π

2
np(1 − p)
^,np(1 − p)P(X = x) ¹ e⁻ 1 ^y2 → 1, where y = √ ^{x − np} .


The convergence is uniform for all x for which y lies in an arbitrary finite inter- val (a, b).

This theorem allows us to approximate the probabilities of the binomial distribu- tion for large n by

= = −

n
∀x; 0 ≤ x ≤ n:
P (X x) p^x(1 p)^n−x x
_≈ 1 ₋ (x − np)²

^√2π np(1 − p) ^exp

P (X = x) ≈ Φ

√

2

— Φ
x − np + ¹
or
2np(1 − p)

√

2

and
x − np − ¹

np(1 − p)
∀x₁, x₂; 0 ≤ x₁ ≤ x₂ ≤ n:

P (x₁ ≤ X ≤ x₂) ≈ Φ

√

2
x₂ − np + ¹



np(1 − p)

— Φ

√

2

,
x₁ − np − ¹

np(1 − p) np(1 − p)
where Φ is the distribution function of the standard normal distribution. The ap- proximation is reasonably good for np(1 − p) > 9.

8. The χ² Distribution

If one forms the sum of m independent, standard normally distributed random vari- ables (expected value 0 and variance 1), one obtains a random variable X with the density function

₀ m

_m

for x < 0,
x

f_X(x; m) =
1
m
2 ² ·Г( ₂ )
· x ^{2 −1} · e^{− 2} for x ≥ 0,

²¹This theorem bears its name in recognition of the French mathematicians Abraham de Moivre (1667–1754) and Pierre-Simon de Laplace (1749–1827).

3. 
   Inferential Statistics 349
   

where Г is the so-called Gamma function (generalization of the factorial)

Г(x) = ∫ ∞ t^x−1e^t dt

₀

for x > 0. This random variable is said to be χ²-distributed with m degrees of free- dom. The expected value and variance are
E(X) = m; D²(X) = 2m.
The χ² distribution plays an important role in the statistical theory of hypothesis testing (see Sect. A.4.3), for example, for independence tests.

9. The Exponential Distribution

A random variable X with the density function

αe

^αx for x > 0,
f_X(x; α) = ⁰ ₋
for x ≤ 0,

with α > 0 is said to be exponentially distributed with parameter α. Its distribution function F_X is

F_X(x; α) = ⁰ ₋
for x ≤ 0,

1 − e
The expected value and variance are
^αx for x > 0.

α

_α2
μ = E(X) = ¹ ; σ ² = D²(X) = ¹ .
The exponential distribution is commonly used to model the durations between the arrivals of people or jobs that enter a queue to wait for service or processing.