UNIT 2: RANDOM VARIABLES

STATISTICAL METHODS FOR BUSINESS
UNIT 2: RANDOM VARIABLES
2.1.- Random variable. Discrete and continuous variables
2.2.- Probability distribution of a random variable
2.3.- Characteristics of a random variable. Expected value and
variance
2.4.- Tchebyshev’s inequality
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for
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UNIT 2. GOALS
• To intuitively understand the concept of random variable
and its relevance in the field of economics.
• To analyse discrete and continuous random variables.
• To compute cumulative probabilities and probabilities of
intervals.
• To be able to compute and interpret the expectation and
the variance of a random variable.
• To apply Chebyshev’s Inequality, and to understand its
relevance in applications.
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STATISTICAL METHODS FOR BUSINESS
UNIT 2: RANDOM VARIABLES
2.1.- Random variables. Discrete
and continuous variables
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EMPIRICAL STUDIES
Under certainty
• No. of branches of a
bank.
• Last year’s profits.
Under uncertainty
• Employment level for next
year.
• Inflation expected for next
month.
Future events
Statistical variables
Non-exhaustive
analyses
Random variables
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RANDOM VARIABLES
Description of a random variable and its
probability distribution
 X “Daily demand of oil stations in a town (thousand
litres)”
 Probability calculus: P(X12), P(6<X15)
 Summary of X (statistical measures)
 Description of the probability distribution of X: Usual
probability models
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Random variables
1
1
Working
(W)
1
P(W)
P(X = 1)
RV (X)
P(NW)
P(X = 0)
Not working
(NW)
0
0
0
Probability
Random experiment
Outcome
Induced
probability
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Classes of random variables
No. of employees in a shop.
No. of customers in a bank office.
Population employed in a sector.
DISCRETE
[Finite or countably
infinite range]
CONTINUOUS
National income of a country.
Inflation level.
Consumption of oil.
[Non-countable
range]
Waiting time in a semaphore.
Earnings in a lottery.
Tariff of a service
HYBRID
(
)
(
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for
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STATISTICAL
VARIABLES
•Discrete
•Continuous
•Hybrid
Values
+
Frequencies
Mean
Variance
RANDOM
VARIABLES ALEATORIAS
VARIABLES
CLASS
DESCRIPTION
CHARACTERISTICS
•Discrete
•Continuous
•Hybrid
Values
+
Probabilities
Expectation
Variance
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for
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STATISTICAL METHODS FOR BUSINESS
UNIT 2: RANDOM VARIABLES
2.2.- Probability distribution of a
random variable
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Discrete random variables
➨ Labor status
X discrete RV
Not working
X=0
P(X=0) = 0.25
Working
X=1
P(X=1) = 0.75
Probability function
P(x)

P : x ∈ℜ → [ 0, 1 ]

P( x)≥0

∑ P ( x i )=1
0.75
0.25
0
i
1
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Variable aleatoria
continua
Continuous
random
variables
X “Profits of a bank’s branch”
p1
p2
L0 L1
L2
L3
L4
L5
... L k
X continuous
Probability density
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Probability density
p1
p2
L0 L1
L2
L3
L4
L5
... Lk
Density =
Intervals of
smaller
length
L0 L1
Probability on the interval
Length of the interval
... L k
Density function
f(x)
x
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•FUNCIÓN DE DENSIDAD
Density function
+
f : x ∈ℜ → ℜ


f ( x)≥0
+∞
∫−∞ f ( x) dx=1
Total area = 1
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DISTRIBUTION FUNCTION
Definition
Meaning
F : x∈ℜ → F ( x)
F ( x)= P( X ≤x)∈ [ 0,1 ]
Probability accumulated
up to value x.
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Cumulative probability
V.A. DISCRETA
Not working ➨ X=0
P(X=0) = 0.25
Working ➨ X=1
P(X=1) = 0.75
0
1
X
{
0 if x<0,
F ( x )= 0. 25 if 0≤x<1,
1 if x≥1 .
i
Distribution function
F ( xi )= ∑ P( x j )
j=1
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•Nº MEDALLAS OBTENIDAS POR UN ATLETA
X “No. of medals won by an athlete”
P(x)
Probability function
0,5
Point probability
0,2
0,1
0
F(x)
1
2
3
Distribution function
1
0,9
Cumulative
probability
0,7
0,5
0
1
2
3
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DISTRIBUTION FUNCTION OF A
CONTINUOUS VARIABLE
x
F ( x)=∫−∞ f (t ) dt
Area = Probability
accumulated up to x
x
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Graphical representations
1
1
F(x)
0
X discrete
F(x)
X continuous
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Distribution function
Properties
1. F non-decreasing monotonic:
2.
x 1 <x 2 ⇒ F ( x 1 )≤ F( x 2 )
Lim x→−∞ F( x)= 0
3. Lim
x→+∞
F( x)= 1
4. F is right continuous:
Lim
+
h→0
F( x+h )= F( x )
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Distribution function
Probability function
DISCRETE
Density function
CONTINUOUS
Random variable
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Relationship between probability function
and distribution function (discrete RVs)
✒
✒
i
F ( xi )= ∑ P( x j )
j=1
P ( X = x i )=F ( x i )−F ( xi−1 )
Relationship between density function and
distribution function (continuous RVs)
✒
✒
x
F ( x)=∫−∞ f (t ) dt
f (x)=F '(x)
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•Ilustración function
Probability
•P(x)
•P(x)
•0.5
P ( X = x i )=F ( x i )−F ( xi−1 )
•0.5
•0.2
•0.1
•0
•1
•2
i
F ( xi )= ∑ P( x j )
j=1
•3
•0.2
•0.1
•0
•1
•2
•3
•1
•2
•3
Distribution function
•F(x)
•F(x)
•1
•0.
9
•0.
•1
•0.
9
•0.
7
•0.
5
7
•0.
5
•0
•1
•2
•3
•0
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Probability of an interval
PROBABILIDADES
P ( a< X ≤b )=P ( X ≤b )− P ( X ≤a )= F ( b )−F ( a )
Continuous case
b
P (a< X ≤b)=∫a f ( x) dx
Area = Probability
of the interval
•a
•b
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Change of variable
X

Y = g(X)
X discrete
 Y discrete
X continuous
 Y discrete
X continuous
 Y continuous
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for
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STATISTICAL METHODS FOR BUSINESS
UNIT 2: RANDOM VARIABLES
2.3.- Characteristics of a random
variable. Expectation and variance
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for
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Characteristics of a random variable
VARIABLES
RANDOM
STATISTICAL
Value Frequency
x1
F(x)
X
f1
k
x2
f2 ̄x =∑ x f
i i
Synthesis
i=1
E ( X )= μ
.....
xk
fk
2
S =( X −̄x )
2
Variance
2
2
σ =E [ X −E ( X )]
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Expectation
Deviation
Random variable X
Random error
X- 
Expected value
E(X)=
E[X- ]=0
The expected deviation around  is null.
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Expected value
X discrete
E ( X )=∫ x dF( x)
X continuous
ℜ
+∞
E ( X )=∑ x i pi
E( X )= ∫ x f ( x) dx
i
−∞
Properties
• E(c) = c
a , c∈ℜ
• E(aX) = a E(X)
• E(X+c) = E(X) + c
• E(X+Y) = E(X)+E(Y)
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Variance
Var ( X )=E [ ( X − μ )
X DISCRETE
2
]
2
Var ( X )=∑ ( x i − μ ) pi
i
+∞
X CONTINUOUS
2
Var ( X )= ∫ ( x− μ ) f ( x ) dx
−∞
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Properties of the variance
Shortened
formula
Var( X )= E ( X ) − μ
• 2  0
2
2
a,c∈ℜ
• Var(X+c) = Var(X)
• Var(aX) = a2 Var(X)
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for
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STATISTICAL METHODS FOR BUSINESS
UNIT 2: RANDOM VARIABLES
2.4.- Tchebyshev’s inequality
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for
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TCHEBYSHEV’S INEQUALITY
∣X −μ∣≥ε
μ−ε
μ
∣X −μ∣<ε
μ+ε
Let X be a RV with finite expectation and variance. Then for any
2
positive constant :
and
σ
P (∣ X − μ∣≥ε )≤ 2
ε
2
σ
P (∣ X − μ∣<ε )≥1− 2
ε
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Tchebyshev’s bounds
Value k
Lower bound for
P(|X-E(X)|<k)
Upper bound for
P(|X-E(X)|k)
1
0
1
2
0.75
0.25
3
0.89
0.11
4
0.9375
0.0625
5
0.96
0.04
10
0.99
0.01
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