14-ma`ruza. Variance and Standard Deviation

The Geometric Distribution

with
Let X be a random variable that describes the number of trials in a Bernoulli exper- iment that are needed until an event A, which occurs with p = P (A) > 0 in each trial, occurs for the first time. Then X has the distribution ∀x ∈ N : (x; P (X = x))
P (X = x) = g_X(x; p) = p(1 − p)^x−1

A.3 Probability Theory 345

and is said to be geometrically distributed with parameter p. In order to compute the probabilities the recursive relation

∀x ∈ N : P (X = x + 1) = (1 − p)P(X = x) with P (X = 1) = p
can be useful. The expected value and variance are
μ = E(X) = ¹ ; σ ² = D²(X) = ^{1 − p}.
p p²

4. The Hypergeometric Distribution

−
From an urn which contains M black and N M white, and thus in total N balls, n balls are drawn without replacement. Let X be the random variable that de- scribes the number of black balls that have been drawn. Then X has the distribution

M N −M
∀x; max(0,n − (N − M)) ≤ x ≤ min(n, M) : (x; P (X = x)) with

x
P (X = x) = h_X(x; n, M, N) =
n−x

_N
n

and is said to be hypergeometrically distributed with parameters n, M and N . This distribution satisfies the recursive relation
∀x; max(0,n − (N − M)) ≤ x ≤ min(n, M) :
h (x + 1; n, M, N) = ^{(M − x)(n − x)} h (x; n, M, N)

_X

X

with h_X

(x + 1)(N − M − n + x + 1)
M (1; n, M, N) = _N .

N
With p = ^M and q = 1 − p, the expected value and variance are

N − 1
μ = E(X) = np; σ ² = D²(X) = npq ^{N − n}.

5. The Poisson Distribution

A random variable X with the distribution ∀x ∈ N : (x; P (X = x)) where

λ _λ
x
P (X = x) = Λ_X(x; λ) = _x! e⁻
is said to be Poisson distributed²⁰ with parameter λ. This distribution satisfies the recursive relation

∀x ∈ N₀ : Λ_X
λ (x + 1; λ) = _{x + 1}
Λ_X(x; λ) with Λ_X
(0; λ) = e^−λ.

²⁰This distribution bears its name in recognition of the French mathematician Siméon-Denis Pois- son (1781–1840).

346 A Statistics

The expected value and variance are

μ = E(X) = λ; σ ² = D²(X) = λ.
A Poisson distribution describes the occurrence frequency of a certain type of events in a fixed duration of time, for example, the number of deadly traffic accidents.

λ _λ
For rare events A and a large number n of trials, the binomial distribution can be approximated by a Poisson distribution, because the following relation holds: if in a binomial distribution, n goes to infinity so that np = λ stays constant, then

n→∞
∀x ∈ N : lim
np=λ
x
b_X(x; p, n) = _x! e⁻ ,

and thus for large n and small p, we obtain the approximation

∀x ∈ N : b_X(x; p, n) ≈
np^x
e
x!
−np_.

For Poisson distributed random variables, the following reproduction law holds:


Theorem A.17 Let X and Y be two (stochastically) independent Poisson-distrib- uted random variables with parameters λ_X and λ_Y , respectively. Then the sum Z = X + Y is also Poisson distributed with parameter λ_Z = λ_X + λ_Y .

6. The Uniform Distribution

A random variable X with the density function

f_X(x; a, b) =
1
b−a
for x ∈ [a, b],

0 otherwise,

∈ [ ]
with a, b R, a < b, is said to be uniformly distributed in a, b . Its distribution function F_X is

x−a b−a
^⎧⎨ ^{0 for x ≤ a,}

F_X(x; a, b) =

^for a ≤ x ≤ b,

The expected value and variance are

^⎩ 1 for x ≥ b.

μ = E(X) =
a + b _{σ 2}

;
2
= D²(X) =
(b a)²

−
12 ^.

7. The Normal Distribution

A random variable X with the density function

²; = √_X
1
N x μ, σ e
2πσ ²
(x μ)² 2σ ²

−
is said to be normally distributed with parameters μ and σ ².

3. 
   Probability Theory 347
   

The expected value and variance are

E(X) = μ; D²(X) = σ ².

= =
The normal distribution with expected value μ 0 and variance σ ² 1 is called standard normal distribution.

= ±

;
The density function f_X(x μ, σ ²) has its maximum at μ and inflection points at x μ σ . The distribution function of X does not possess a closed-form represen- tation. As a consequence, it is usually tabulated, most commonly for the standard normal distribution, from which the values of arbitrary normal distributions can be easily obtained by simple linear transformations.
However, in practice one often faces the reversed problem, namely to find the argument of the distribution function of the standard normal distribution for which is has a given value (or, in other words, one desires to find a quantile of the normal distribution). In order to solve this problem one may just as well use tabulated val- ues. However, the inverse function can be approximated fairly well by the ratio of two polynomials, which is usually employed in computer programs (e.g., [15]).
The normal distribution is certainly the most important continuous distribution, since many random processes, especially measurements of physical quantities, can be described well by this distribution. The theoretical justification for this observed fact is the important central limit theorem:



i
Theorem A.18 (central limit theorem) Let X₁,..., X_m be m (stochastically) in- dependent real-valued random variables. In addition, let them satisfy the so-called Lindeberg condition, that is, if F_i(x) are the distribution functions of the random variables X_i , i = 1,..., m, μ_i their expected values, and σ ² their variances, then

for every ε > 0, it is

lim
m→∞

m



1
_V 2
^m i=1


(x_i μ_i)²

−

m

∫
|x_i −μ_i |>εV ²

dF_i(x) = 0

with V ² = ^.m σ ². Then the standardized sums

i=1
^{m i=1} i
^.m (X_i − μ_i)

S_m = ,_{.m 2}
^{i=1 σ}i
(that is, standardized to expected value 0 and variance 1) satisfy

∀x ∈ R: lim
1 ^x

∫
P (S_m ≤ x) = Φ(x) = √ e

2

_— t
² dt,

m→∞
^2π −∞

where Φ(x) is the distribution function of the standard normal distribution.

Intuitively this theorem says that the sum of a large number of almost arbitrarily distributed random variables (the Lindeberg condition is a very weak restriction) is approximately normally distributed. Since physical measurements are usually af- fected by a large number of random influences from several independent sources,

348 A Statistics

which all add up to form the total measurement error, the result is often approxi- mately normally distributed. The central limit theorem thus explains why normally distributed quantities are so common in practice.

Like the Poisson distribution, the normal distribution can be used as an approxi- mation of the binomial distribution, even if the probabilities p are not small.