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Statistics 305
Some Useful Populations Defined
in Mathematical Theory
(Supplemental Material)
Populations defined in mathematical theory do not exist, they are useful figments of our
imagination. Random samples from such theoretical populations can be taken using numerical
algorithms available in programs like JMP.
The populations described below are all theoretical and are grouped into families having one or
more indexing parameters to identify family members. The elements of every population are
numbers. The set of unique elements in some of the populations is a finite or countably infinite
set of numbers. Other populations are such that the unique elements form a continuum. Each
population has an associated function, called the Cumulative Distribution Function (CDF),
which can be evaluated to find the proportion of population elements which are less than or
equal to any given number. A second function, called the Density Function, is also associated
with each population. Its graph shows how population proportions change across the unique real
number population elements. When we study Probability in Chapter 5, these will be called
Probability functions, and proportions will be called probabilities. Don’t worry about probability
now. The general term Discrete Population is used for any population whose unique elements
form a finite or countably infinite set. Continuous Populations have unique elements that form a
continuum. Every theoretical population, discrete or continuous, will be assumed to have an
infinite number of elements.
1. Binomial Family of Populations B(n, p). (These are Discrete Populations.)
Indexing parameters: n (an integer > 0) and p (a real number 0 < p < 1).
Unique population elements: {0, 1, 2, …, n}.
Density Function:
n!
⎧
x
n− x
if x = 0, 1, 2, ..., n
⎪⎪ x ! (n − x) ! p (1 − p )
f ( x) = ⎨
⎪
⎪⎩ 0 otherwise
Cumulative Distribution Function for any x = 0, 1, 2, …, n
F ( x) =
x
∑
n
f ( y)
(Note that
y =0
∑
f ( y) = 1 )
y =0
y integer
y integer
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Example: A Binomial B(3, 0.1) is any population whose unique elements are 0, 1, 2
3!
3!
and 3 and these occur in respective proportions
(0.1)0(0.9)3,
(0.1) (0.9)2,
0 !3!
1! 2 !
3!
3!
(0.1)2 (0.9) and
(0.1)3 which are density function values. Evaluating these
2 !1 !
3!0!
expressions we have 0.729, 0.243, 0.027, and 0.001. Thus almost 73% of the elements
in a Binomial B (3, 0.1) population are zero. The Cumulative Distribution Function
value at 2 is F(2) = 0.729 + 0.243 + 0.027 = 0.999. Thus 99.9% of the elements in a
Binomial (3, 0.1) population do not exceed 2 (i.e. are not 3). The graph of the Density
Function is a bar chart having 4 bars. (See Figure 1 for plots of B (5, 0.2), B (10, 0.2)
… etc).
2. Geometric Family of Populations G(p). (These are Discrete Populations.)
Indexing parameter: p (any real number 0 < p < 1).
Unique population elements: {all the positive integers}
Density function:
⎧ p(1 − p) x −1 for x = 1, 2, ...
⎪
f ( x) = ⎨
⎪ 0 otherwise
⎩
Cumulative Distribution Function for an integer x > 0
F ( x) =
x
∑
∞
p (1 − p ) y −1
(Note:
y =1
∑
f ( y) = 1 )
y =1
y integer
See Figure 2 for a graph of the Density Function.
3. Normal Family of Populations N(μ, σ 2). (These are Continuous Populations.)
Indexing parameters: μ (any finite real number)
σ 2 (any finite positive real number).
Unique population elements: all real numbers (-∞, ∞).
Density function:
φ ( x) =
1
σ 2Π
−
e
1 ⎛ x−μ ⎞
⎜
⎟
2⎝ σ ⎠
2
−∞ < x < ∞
Cumulative Distribution Function:
x
Φ( x) =
∫
φ ( y )dy
for any x, − ∞ < x < ∞
−∞
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∞
∫
( Note:
φ ( y )dy = 1 )
−∞
The graph of the Density Function is a bell-shaped curve, as shown in Figure 3. The
Standard Normal Population is that one indexed by μ = 0 and σ 2 = 1, i.e. N(0, 1).
4. Exponential Family of Populations E (α). (These are Continuous Populations.)
Indexing parameter: α (any real number > 0)
Unique population elements: (all real numbers > 0)
Density function:
⎧ 1 −
⎪
⎪α e
f ( x) = ⎨
⎪
⎪⎩ 0
x
α
for x > 0
otherwise
Cumulative Distribution Function:
⎧0
⎪
⎪
F ( x) = ⎨ x
−
⎪ f ( y )dy = 1 − e
∫
⎪
⎩0
if x ≤ 0
x
α
if x > 0
Figure 4 shows graphs of Exponential Density Functions. The Standard Exponential
Population is E(1).
5. Uniform Family of Populations. U(a, b). (These are Continuous Populations.)
Indexing parameters: interval endpoints (a, b). (a and b are numbers with a < b.)
Unique population elements: all real numbers in the interval (a, b).
Density function:
⎧ 1
⎪⎪
f ( x) = ⎨ b − a
⎪
⎩⎪ 0
a< x<b
otherwise
Cumulative Distribution Function:
x
F ( x) =
∫
f ( y )dy =
−∞
x−a
,
b−a
The Standard Uniform Population is U(0, 1).
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a< x<b
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