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Random Variable
Random VariablesSTATISTICS – Lecture no. 3
Jirı Neubauer
Department of Econometrics FEM UO Brnooffice 69a, tel. 973 442029email:[email protected]
13. 10. 2009
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Random Variable
Many random experiments have numerical outcomes.
Definition
A random variable is a real-valued function X (ω) defined on thesample space Ω.
The set of possible values of the random variable X is called therange of X .
M = x ;X (ω) = x.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Random Variable
We denote random variables by capital letters X ,Y , . . .(eventually X1,X2, . . . ) and their particular values by smallletters x , y , . . . . Using random variables we can describerandom events, for example X = x ,X ≤ x , x1 < X < x2 etc.
Examples of random variables:
the number of dots when a die is rolled, the range isM = 1, 2, . . . 6the number of rolls of a die until the first 6 appears, the rangeis M = 1, 2, the lifetime of the lightbulb, the range is M = x ; x ≥ 0,
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Random Variable
We denote random variables by capital letters X ,Y , . . .(eventually X1,X2, . . . ) and their particular values by smallletters x , y , . . . . Using random variables we can describerandom events, for example X = x ,X ≤ x , x1 < X < x2 etc.
Examples of random variables:
the number of dots when a die is rolled, the range isM = 1, 2, . . . 6the number of rolls of a die until the first 6 appears, the rangeis M = 1, 2, the lifetime of the lightbulb, the range is M = x ; x ≥ 0,
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Random Variable
According to the range M we separate random variables to
discrete . . .M is finite or countable,
continuous . . .M is a closed or open interval.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Random Variable
According to the range M we separate random variables to
discrete . . .M is finite or countable,
continuous . . .M is a closed or open interval.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Random Variable
Examples of discrete random variables:
the number of cars sold at a dealership during a given month,M = 0, 1, 2, . . . the number of houses in a certain block, M = 1, 2, . . . the number of fish caught on a fishing trip, M = 0, 1, 2, . . . the number of heads obtained in three tosses of a coin,M = 0, 1, 2, 3
Examples of continuous random variables:
the height of a person, M = (0,∞)
the time taken to complete an examination, M = (0,∞)
the amount of milk in a bottle, M = (0,∞)
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Random Variable
Examples of discrete random variables:
the number of cars sold at a dealership during a given month,M = 0, 1, 2, . . . the number of houses in a certain block, M = 1, 2, . . . the number of fish caught on a fishing trip, M = 0, 1, 2, . . . the number of heads obtained in three tosses of a coin,M = 0, 1, 2, 3
Examples of continuous random variables:
the height of a person, M = (0,∞)
the time taken to complete an examination, M = (0,∞)
the amount of milk in a bottle, M = (0,∞)
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Random Variable
For the description of random variables we will use some functions:
a (cumulative) distribution function F (x),
a probability function p(x) – only for discrete randomvariables,
a probability density function f (x) – only for continuousrandom variables.
and some measures:
measures of location,
measures of dispersion,
measures of concentration.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Random Variable
For the description of random variables we will use some functions:
a (cumulative) distribution function F (x),
a probability function p(x) – only for discrete randomvariables,
a probability density function f (x) – only for continuousrandom variables.
and some measures:
measures of location,
measures of dispersion,
measures of concentration.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Distribution Function
Definition
Let X be any random variable. The distribution function F (x) ofthe random variable X is defined as
F (x) = P(X ≤ x), x ∈ R.
Note: distribution function = cumulative distribution function
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Distribution Function
We mention some important properties of F (x):
for every real x : 0 ≤ F (x) ≤ 1,
F (x) is a non-decreasing, right-continuous function,
it has limits
limx→−∞
F (x) = 0, limx→∞
F (x) = 1,
if range of X is M = x ; x ∈ (a, b〉 then F (a) = 0 aF (b) = 1,
for every real numbers x1 and x2:P(x1 < X ≤ x2) = F (x2)− F (x1).
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Distribution Function
We mention some important properties of F (x):
for every real x : 0 ≤ F (x) ≤ 1,
F (x) is a non-decreasing, right-continuous function,
it has limits
limx→−∞
F (x) = 0, limx→∞
F (x) = 1,
if range of X is M = x ; x ∈ (a, b〉 then F (a) = 0 aF (b) = 1,
for every real numbers x1 and x2:P(x1 < X ≤ x2) = F (x2)− F (x1).
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Distribution Function
We mention some important properties of F (x):
for every real x : 0 ≤ F (x) ≤ 1,
F (x) is a non-decreasing, right-continuous function,
it has limits
limx→−∞
F (x) = 0, limx→∞
F (x) = 1,
if range of X is M = x ; x ∈ (a, b〉 then F (a) = 0 aF (b) = 1,
for every real numbers x1 and x2:P(x1 < X ≤ x2) = F (x2)− F (x1).
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Distribution Function
We mention some important properties of F (x):
for every real x : 0 ≤ F (x) ≤ 1,
F (x) is a non-decreasing, right-continuous function,
it has limits
limx→−∞
F (x) = 0, limx→∞
F (x) = 1,
if range of X is M = x ; x ∈ (a, b〉 then F (a) = 0 aF (b) = 1,
for every real numbers x1 and x2:P(x1 < X ≤ x2) = F (x2)− F (x1).
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Discrete Random Variable
For a discrete random variable X , we are interested in computingprobabilities of the type P(X = xk) for various values xk in rangeof X .
Definition
Let X be a discrete random variable with range x1, x2, . . . (finiteor countably infinite). The function
p(x) = P(X = x)
is called the probability function of X .
Note: probability function = probability mass function
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Discrete Random Variable
We mention some important properties of p(x)
for every real number x , 0 ≤ p(x) ≤ 1,
∑x∈M
p(x) = 1
for every two real numbers xk and xl (xk ≤ xl):
P(xk ≤ X ≤ xl) =
xl∑xi=xk
p(xi ).
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Discrete Random Variable
We mention some important properties of p(x)
for every real number x , 0 ≤ p(x) ≤ 1,∑x∈M
p(x) = 1
for every two real numbers xk and xl (xk ≤ xl):
P(xk ≤ X ≤ xl) =
xl∑xi=xk
p(xi ).
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Discrete Random Variable
We mention some important properties of p(x)
for every real number x , 0 ≤ p(x) ≤ 1,∑x∈M
p(x) = 1
for every two real numbers xk and xl (xk ≤ xl):
P(xk ≤ X ≤ xl) =
xl∑xi=xk
p(xi ).
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Probability Function
The probability function p(x) can be described by
the table,
X x1 x2 . . . xi . . .∑
p(x) p(x1) p(x2) . . . p(xi ) . . . 1
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Probability Function
the graph [x , p(x)],
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Probability Function
the formula, for example
p(x) =
π(1− π)x x = 0, 1, 2, . . . ,
0 otherwise,
where π is a given probability.
Jirı Neubauer Random Variables
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Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
The shooter has 3 bullets and shoots at the target until the firsthit or until the last bullet. The probability that the shooter hits thetarget after one shot is 0.6. The random variable X is the numberof the fired bullets. Find the probability and the distributionfunction of the given random variable. What is the probability thatthe number of the fired bullets will not be larger then 2?
Jirı Neubauer Random Variables
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Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
Random variable X is discrete with the range M = 1, 2, 3. Theprobability function is:
p(1) = P(X = 1) = 0.6,
p(2) = P(X = 2) = 0.4 · 0.6 = 0.24,
p(3) = P(X = 3) = 0.4·0.4·0.6+0.4·0.4·0.4 = 0.4·0.4 = 0.16.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
Random variable X is discrete with the range M = 1, 2, 3. Theprobability function is:
p(1) = P(X = 1) = 0.6,
p(2) = P(X = 2) = 0.4 · 0.6 = 0.24,
p(3) = P(X = 3) = 0.4·0.4·0.6+0.4·0.4·0.4 = 0.4·0.4 = 0.16.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
Random variable X is discrete with the range M = 1, 2, 3. Theprobability function is:
p(1) = P(X = 1) = 0.6,
p(2) = P(X = 2) = 0.4 · 0.6 = 0.24,
p(3) = P(X = 3) = 0.4·0.4·0.6+0.4·0.4·0.4 = 0.4·0.4 = 0.16.
Jirı Neubauer Random Variables
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Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
All results are summarized in the table
x 1 2 3∑
p(x) 0.6 0.24 0.16 1
The probability function can be described by the formula
p(x) =
0.6 · 0.4x−1 x = 1, 2,
0.42 x = 3,0 otherwise.
Jirı Neubauer Random Variables
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Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
We can calculate some values of the distribution function F (x):
F (0) = P(X ≤ 0) = 0,
F (1) = P(X ≤ 1) = p(1) = 0.6,
F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6,
F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84,
F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1,
F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
We can calculate some values of the distribution function F (x):
F (0) = P(X ≤ 0) = 0,
F (1) = P(X ≤ 1) = p(1) = 0.6,
F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6,
F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84,
F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1,
F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
We can calculate some values of the distribution function F (x):
F (0) = P(X ≤ 0) = 0,
F (1) = P(X ≤ 1) = p(1) = 0.6,
F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6,
F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84,
F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1,
F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
We can calculate some values of the distribution function F (x):
F (0) = P(X ≤ 0) = 0,
F (1) = P(X ≤ 1) = p(1) = 0.6,
F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6,
F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84,
F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1,
F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
We can calculate some values of the distribution function F (x):
F (0) = P(X ≤ 0) = 0,
F (1) = P(X ≤ 1) = p(1) = 0.6,
F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6,
F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84,
F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1,
F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
We can calculate some values of the distribution function F (x):
F (0) = P(X ≤ 0) = 0,
F (1) = P(X ≤ 1) = p(1) = 0.6,
F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6,
F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84,
F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1,
F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
We can write
F (x) =
0 x < 1,
0.6 1 ≤ x < 2,0.84 2 ≤ x < 3,1 x ≥ 3.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
Figure: The probability and the distribution function
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
What is the probability that the number of the fired bullets will notbe larger then 2?
P(X ≤ 2) = P(X = 1) + P(X = 2) = p(1) + p(2) = F (2) == 0.6 + 0.24 = 0.84.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Probability Density Function
If the cumulative distribution function is a continuous function,then X is said to be a continuous random variable.
Definition
The probability density function of the random variable X is anon-negative function f (x) such that
F (x) =
x∫−∞
f (t)dt, x ∈ R.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Probability Density Function
Some properties of f (x):∞∫−∞
f (x)dx =∫M
f (x)dx = 1,
f (x) = dF (x)dx = F ′(x), where the derivative exists,
P(x1 ≤ X ≤ x2) = P(x1 < X < x2) = P(x1 < X ≤ x2) =
P(x1 ≤ X < x2) = F (x2)− F (x1) =x2∫x1
f (x)dx
If X is a continuous random variable, then P(X = x) = 0.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Probability Density Function
Some properties of f (x):∞∫−∞
f (x)dx =∫M
f (x)dx = 1,
f (x) = dF (x)dx = F ′(x), where the derivative exists,
P(x1 ≤ X ≤ x2) = P(x1 < X < x2) = P(x1 < X ≤ x2) =
P(x1 ≤ X < x2) = F (x2)− F (x1) =x2∫x1
f (x)dx
If X is a continuous random variable, then P(X = x) = 0.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Probability Density Function
Some properties of f (x):∞∫−∞
f (x)dx =∫M
f (x)dx = 1,
f (x) = dF (x)dx = F ′(x), where the derivative exists,
P(x1 ≤ X ≤ x2) = P(x1 < X < x2) = P(x1 < X ≤ x2) =
P(x1 ≤ X < x2) = F (x2)− F (x1) =x2∫x1
f (x)dx
If X is a continuous random variable, then P(X = x) = 0.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Probability Density Function
Some properties of f (x):∞∫−∞
f (x)dx =∫M
f (x)dx = 1,
f (x) = dF (x)dx = F ′(x), where the derivative exists,
P(x1 ≤ X ≤ x2) = P(x1 < X < x2) = P(x1 < X ≤ x2) =
P(x1 ≤ X < x2) = F (x2)− F (x1) =x2∫x1
f (x)dx
If X is a continuous random variable, then P(X = x) = 0.
Jirı Neubauer Random Variables
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Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Probability Density Function
The function f (x) we can describe by a formula or a graph, forexample
f (x) =
15e−
x−25 pro x > 2,
0 pro x ≤ 2.
Jirı Neubauer Random Variables
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Example
The random variable X has the probability density function
f (x) =
cx2(1− x) 0 < x < 1,
0 otherwise.
Determine a constant c in order that f (x) is a probability densityfunction. Find a distribution function of the random variable X .Calculate the probability P(0.2 < X < 0.8).
Jirı Neubauer Random Variables
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Example
The probability density function has to fulfil∫M
f (x)dx = 1.
1∫0
cx2(1− x)dx = c1∫0
(x2 − x3)dx = c[
x3
3 −x4
4
]1
0=
= c[
13 −
14
]= c
12 = 1,
we get c = 12.
Jirı Neubauer Random Variables
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Example
The distribution function can be calculated by the definition ofthe probability density function. We can write for 0 < x < 1
F (x) =x∫0
12t2(1− t)dt = 12x∫0
(t2 − t3)dt = 12[
t3
3 −t4
4
]x
0=
= 12[
x3
3 −x4
4
]= 4x3 − 3x4.
F (x) =
0 x ≤ 0,
x3(4− 3x) 0 < x < 1,1 otherwise.
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Figure: The probability density function and the distribution function
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Example
Using the probability density function we can calculate
P(0.2 < X < 0.8) =
0.8∫0.2
12x2(1− x)dx =[4x3 − 3x4
]0.8
0.2= 0.792.
If the distribution function is known, we can do simpler calculation
P(0.2 < X < 0.8) = F (0.8)− F (0.2) == 0.83(4− 3 · 0.8)− 0.23(4− 3 · 0.2) = 0.792.
Jirı Neubauer Random Variables
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Example
Using the probability density function we can calculate
P(0.2 < X < 0.8) =
0.8∫0.2
12x2(1− x)dx =[4x3 − 3x4
]0.8
0.2= 0.792.
If the distribution function is known, we can do simpler calculation
P(0.2 < X < 0.8) = F (0.8)− F (0.2) == 0.83(4− 3 · 0.8)− 0.23(4− 3 · 0.2) = 0.792.
Jirı Neubauer Random Variables
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Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
A random variable X is described by the distribution function
F (x) =
0 x ≤ 0,
1− e−x x > 0.
Find a probability density function.
Jirı Neubauer Random Variables
Random Variable
Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
Using mentioned formula
f (x) =dF (x)
dx
and the fact that ddx (1− e−x) = e−x we get
f (x) =
0 x ≤ 0,
e−x x > 0.
Jirı Neubauer Random Variables
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Measures of Location
The distribution function (the probability function or theprobability density function) gives us the complete informationabout the random variable. Sometimes is useful to know somesimpler and concentrated formulation of this information such asmeasures of location, dispersion and concentration.
The best known measures of location are a mean (an expectedvalue), quantiles (a median, an upper and a lower quartile, . . . )and a mode.
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Expected Value
Definition
The mean (the expected value) E (X ) of the random variable X(sometimes denoted as µ) is the value that is expected to occurper repetition, if an experiment is repeated a large number oftimes. For the discrete random variable is defined as
E (X ) =∑M
xip(xi ),
for the continuous random variable as
E (X ) =
∫M
xf (x)dx
if the given sequence or integral absolutely converges.Jirı Neubauer Random Variables
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Expected Value
We mention some properties of the mean:
the mean of the constant c is equal to this constant
E (c) = c ,
the mean of the product of the constant c and the randomvariable X is equal to the product of the given constant c andthe mean of X
E (cX ) = cE (X ),
Jirı Neubauer Random Variables
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Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Expected Value
We mention some properties of the mean:
the mean of the constant c is equal to this constant
E (c) = c ,
the mean of the product of the constant c and the randomvariable X is equal to the product of the given constant c andthe mean of X
E (cX ) = cE (X ),
Jirı Neubauer Random Variables
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Expected Value
the mean of the sum of random variables X1,X2, . . . ,Xn isequal to the sum of the mean of the given random variables,
E (X1 + X2 + · · ·+ Xn) = E (X1) + E (X2) + · · ·+ E (Xn),
if X1,X2, . . . ,Xn are independent, then the mean of theirproduct is equal to the product of their means
E (X1X2 . . .Xn) = E (X1)E (X2) . . .E (Xn).
Jirı Neubauer Random Variables
Random Variable
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Expected Value
the mean of the sum of random variables X1,X2, . . . ,Xn isequal to the sum of the mean of the given random variables,
E (X1 + X2 + · · ·+ Xn) = E (X1) + E (X2) + · · ·+ E (Xn),
if X1,X2, . . . ,Xn are independent, then the mean of theirproduct is equal to the product of their means
E (X1X2 . . .Xn) = E (X1)E (X2) . . .E (Xn).
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Random Variables and Independency
The random variables X1,X2, . . . ,Xn are independent if and only iffor any numbers x1, x2, . . . , xn ∈ R is
P(X1 ≤ x1,X2 ≤ x2, . . . ,Xn ≤ xn) =
= P(X1 ≤ x1) · P(X2 ≤ x2) · · ·P(Xn ≤ xn).
Let us have the random vector X = (X1,X2, . . . ,Xn), whosecomponents X1,X2, . . . ,Xn are the random variables.F (x) = F (x1, x2, . . . , xn) = P(X1 ≤ x1,X2 ≤ x2, . . . ,Xn ≤ xn) isthe distribution function of the vector X andF (x1),F (x2), . . . ,F (xn) are the distribution functions of therandom variables X1,X2, . . . ,Xn. The random variablesX1,X2, . . . ,Xn are independent if and only if
F (x1, x2, . . . , xn) = F (x1) · F (x2) · · ·F (xn).
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Random Variables and Independency
If X is the random vector whose components are the discreterandom variables, the functionp(x) = p(x1, x2, . . . , xn) = P(X1 = x1,X2 = x2, . . . ,Xn = xn) is theprobability function of the vector X, p(x1), p(x2), . . . , p(xn) are theprobability functions of X1,X2, . . . ,Xn, then:X1,X2, . . . ,Xn are independent if and only if
p(x1, x2, . . . , xn) = p(x1) · p(x2) · · · p(xn).
If X is the random vector whose components are the continuousrandom variables, the function f (x) = f (x1, x2, . . . , xn) is theprobability density function of the vector X, f (x1), f (x2), . . . , f (xn)are the probability density functions of X1,X2, . . . ,Xn, then:X1,X2, . . . ,Xn are independent if and only if
f (x1, x2, . . . , xn) = f (x1) · f (x2) · · · f (xn).
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Quantile
Definition
100P% quantile xP of the random variable with the increasingdistribution function is such value of the random variable that
P(X ≤ xP) = F (xP) = P, 0 < P < 1.
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Quantile
The quantile x0.50 we call median Me(X ), it fulfilsP(X ≤ Me(X )) = P(X ≥ Me(X )) = 0.50. The quantile x0.25 iscalled the lower quartile, the quantile x0.75 is called the upperquartile. The selected quantiles of some important distributionsare tabulated.
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Mode
Definition
The mode Mo(X ) is the value of the random variable with thehighest probability (for the discrete random variable), or the value,where the function f (x) has the maximum (for the continuousrandom variable ).
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Example
Find the mean (the expected value) and the mode of the randomvariable defined as the number of fired bullets (see the examplebefore). The probability function is
p(x) =
0.6 · 0.4x−1 x = 1, 2,
0.42 x = 3,0 otherwise.
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Example
The mean (the expected value) we get using the formula from thedefinition of E (X )
E (x) =3∑
i=1
xip(xi ) = 1 · 0.6 · 0.40 + 2 · 0.6 · 0.41 + 3 · 0.42 = 1.56.
The mode is the value of the given random variable with thehighest probability which is Mo(X ) = 1, because p(1) = 0.6.
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Example
The mean (the expected value) we get using the formula from thedefinition of E (X )
E (x) =3∑
i=1
xip(xi ) = 1 · 0.6 · 0.40 + 2 · 0.6 · 0.41 + 3 · 0.42 = 1.56.
The mode is the value of the given random variable with thehighest probability which is Mo(X ) = 1, because p(1) = 0.6.
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Example
The random variable X is described by the probability densityfunction
f (x) =
12x2(1− x) 0 < x < 1,
0 otherwise.
Find the mean (the expected value) and the mode.
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Example
We calculate the mean using the definition formula
E (X ) =
1∫0
xf (x)dx =
1∫0
x ·12x2(1−x) = 12
[x4
4− x5
5
]1
0
=3
5= 0.6.
The mode is the maximum of the probability density function. Wehave to find the maximum of f (x) on the interval 0 < x < 1,ddx
[12x2(1− x)
]= 12(2x − 3x2) = 0, x(2− 3x) = 0, we get x = 0
or x = 2/3. The maximum of f (x) is in x = 2/3 ⇒ Mo(X ) = 2/3.
Jirı Neubauer Random Variables
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Example
We calculate the mean using the definition formula
E (X ) =
1∫0
xf (x)dx =
1∫0
x ·12x2(1−x) = 12
[x4
4− x5
5
]1
0
=3
5= 0.6.
The mode is the maximum of the probability density function. Wehave to find the maximum of f (x) on the interval 0 < x < 1,ddx
[12x2(1− x)
]= 12(2x − 3x2) = 0, x(2− 3x) = 0, we get x = 0
or x = 2/3. The maximum of f (x) is in x = 2/3 ⇒ Mo(X ) = 2/3.
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Example
Find the median,the upper and the lower quartile of the randomvariable X with the distribution function
F (x) =
1− 1
x3 x > 1,0 x ≤ 1.
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Example
The quantile is defined by the formula F (xP) = P.
1− 1
x3P
= P,
xP =1
3√
1− P.
median x0.50 = 13√1−0.50
= 1.260,
lower quartile x0.25 = 13√1−0.25
= 1.101,
upper quartile x0.75 = 13√1−0.75
= 1.587.
Jirı Neubauer Random Variables
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Example
The quantile is defined by the formula F (xP) = P.
1− 1
x3P
= P,
xP =1
3√
1− P.
median x0.50 = 13√1−0.50
= 1.260,
lower quartile x0.25 = 13√1−0.25
= 1.101,
upper quartile x0.75 = 13√1−0.75
= 1.587.
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Measures of Dispersion
The elementary and widely-used measures of dispersion are thevariance and the standard deviation.
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Variance
Definition
The variance D(X ) of the random variable X (sometimes denotedas σ2) is defined by the formula
D(X ) = E[X − E (X )]2
.
The variance of the discrete random variable is given by
D(X ) =∑M
[xi − E (X )]2p(xi ),
the variance of the continuous random variable is
D(X ) =
∫M
[x − E (X )]2f (x)dx .
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Variance
Some properties of the variance:
D(k) = 0, where k is a constant,
D(kX ) = k2D(X ),
D(X + Y ) = D(X ) + D(Y ), if X and Y are independent,
D(X ) ≥ 0 for every random variable,
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Variance
Some properties of the variance:
D(k) = 0, where k is a constant,
D(kX ) = k2D(X ),
D(X + Y ) = D(X ) + D(Y ), if X and Y are independent,
D(X ) ≥ 0 for every random variable,
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Variance
Some properties of the variance:
D(k) = 0, where k is a constant,
D(kX ) = k2D(X ),
D(X + Y ) = D(X ) + D(Y ), if X and Y are independent,
D(X ) ≥ 0 for every random variable,
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Variance
Some properties of the variance:
D(k) = 0, where k is a constant,
D(kX ) = k2D(X ),
D(X + Y ) = D(X ) + D(Y ), if X and Y are independent,
D(X ) ≥ 0 for every random variable,
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Variance
D(X ) = E (X 2)− E (X )2,D(X ) = E [X − E (X )]2 = E [X 2 − 2XE (X ) + E (X )2] =
= E (X 2)− E [2XE (X )] + E [E (X )2] == E (X 2)− 2E (X )E (X ) + E (X )2 = E (X 2)− E (X )2
For the discrete random variable
D(X ) =∑M
x2i p(xi )− E (X )2,
for the continuous random variable
D(X ) =
∫M
x2f (x)dx − E (X )2.
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Standard Deviation
Definition
The standard deviation σ(X ) of the random variable X isdefined as the square root of the variance
σ(X ) =√
D(X ).
The standard deviation has the same unit as the random variableX .
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Figure: Relation between the mean and the standard deviation
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Example
Find the variance and the standard deviation of the randomvariable defined as the number of fired bullets (see the examplebefore).
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Example
The mean is E (X ) = 1.56 (see the previous example). For thepurpose of calculation of the variance we use the formula
D(X ) = E (X 2)− E (X )2
E (X 2) =∑M
x2i p(xi ) =
3∑i=1
x2i p(xi ) = 12·0.6+22·0.24+32·0.16 = 3,
thenD(X ) = 3− 1.562 = 0.566.
The standard deviation is the square root of the variance
σ(X ) =√
D(X ) = 0.753.
Jirı Neubauer Random Variables
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Example
The mean is E (X ) = 1.56 (see the previous example). For thepurpose of calculation of the variance we use the formula
D(X ) = E (X 2)− E (X )2
E (X 2) =∑M
x2i p(xi ) =
3∑i=1
x2i p(xi ) = 12·0.6+22·0.24+32·0.16 = 3,
thenD(X ) = 3− 1.562 = 0.566.
The standard deviation is the square root of the variance
σ(X ) =√
D(X ) = 0.753.
Jirı Neubauer Random Variables
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Example
The mean is E (X ) = 1.56 (see the previous example). For thepurpose of calculation of the variance we use the formula
D(X ) = E (X 2)− E (X )2
E (X 2) =∑M
x2i p(xi ) =
3∑i=1
x2i p(xi ) = 12·0.6+22·0.24+32·0.16 = 3,
thenD(X ) = 3− 1.562 = 0.566.
The standard deviation is the square root of the variance
σ(X ) =√
D(X ) = 0.753.
Jirı Neubauer Random Variables
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Discrete Random VariableContinuous Random VariableMeasures of LocationMeasures of DispersionMeasures of Concentration
Example
Find the variance and the standard deviation of the randomvariable X with the probability density function
f (x) =
12x2(1− x) 0 < x < 1,
0 otherwise.
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Example
The mean is E (X ) = 3/5 (see the previous example).
D(X ) = E (X 2)− E (X )2
E (X 2) =∫M
x2f (x)dx =1∫0
x2 · 12x2(1− x)dx = 12[
x5
5 −x6
6
]1
0
= 25 = 0.4,
D(X ) =2
5−
(3
5
)2
=1
25= 0.04.
The standard deviation is
σ(X ) =√
D(X ) =1
5= 0.2.
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Example
The mean is E (X ) = 3/5 (see the previous example).
D(X ) = E (X 2)− E (X )2
E (X 2) =∫M
x2f (x)dx =1∫0
x2 · 12x2(1− x)dx = 12[
x5
5 −x6
6
]1
0
= 25 = 0.4,
D(X ) =2
5−
(3
5
)2
=1
25= 0.04.
The standard deviation is
σ(X ) =√
D(X ) =1
5= 0.2.
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Measures of Concentration
We will focus on the measures describing the shape of randomvariables distribution (skewness and kurtosis). These measures aredefined by moments.
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Measures of Concentration
Definition
The r th moment µ′r of the random variable X is defined by theformula
µ′r (X ) = E (X r ) for r = 1, 2, . . . .
The r th moment of the discrete random variable is given by
µ′r (X ) =∑M
x ri p(xi ),
r th moment of the continuous random variable is
µ′r (X ) =
∫M
x r f (x)dx .
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Measures of Concentration
Definition
The r th central moment µr of the random variable X is definedby the formula
µr (X ) = E [X − E (X )]r for r = 2, 3, . . . .
The r th central moment of the discrete random variable is given by
µr (X ) =∑M
[xi − E (X )]rp(xi ),
the r th central moment of the continuous random variable is
µr (X ) =
∫M
[x − E (X )]r f (x)dx .
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Skewness
Definition
The skewness α3(X ) is defined by the formula
α3(X ) =µ3(X )
σ(X )3.
According to the values of the skewness we can tell whetherdistribution is symmetric or asymmetric.
α3 = 0, distribution is symmetric,
α3 < 0, distribution is skewed to the right,
α3 > 0, distribution is skewed to the left.
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Skewness
Definition
The skewness α3(X ) is defined by the formula
α3(X ) =µ3(X )
σ(X )3.
According to the values of the skewness we can tell whetherdistribution is symmetric or asymmetric.
α3 = 0, distribution is symmetric,
α3 < 0, distribution is skewed to the right,
α3 > 0, distribution is skewed to the left.
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Skewness
Figure: The skewnessJirı Neubauer Random Variables
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Kurtosis
Definition
The kurtosis α4(X ) is defined by the formula
α4(X ) =µ4(X )
σ(X )4− 3.
Kurtosis is a measure of how outlier-prone a distribution is. Thekurtosis of the normal distribution is 0. Distributions that are moreoutlier-prone than the normal distribution have kurtosis greaterthan 0; distributions that are less outlier-prone have kurtosis lessthan 0.
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Kurtosis
Definition
The kurtosis α4(X ) is defined by the formula
α4(X ) =µ4(X )
σ(X )4− 3.
Kurtosis is a measure of how outlier-prone a distribution is. Thekurtosis of the normal distribution is 0. Distributions that are moreoutlier-prone than the normal distribution have kurtosis greaterthan 0; distributions that are less outlier-prone have kurtosis lessthan 0.
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Kurtosis
Figure: The kurtosisJirı Neubauer Random Variables
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Example
Calculate the skewness and the kurtosis of the random variabledefined as the number of fired bullets (see the previous examples).
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Example
First of all we have to calculate 3rd and 4th central moment. Themean of the given random variable is E (X ) = 1.56, the standarddeviation is σ(X ) = 0.753.
µ3 =3∑
i=1[xi − E (X )]3p(xi ) = (1− 1.56)3 · 0.6+
+(2− 1.56)3 · 0.24 + (3− 1.56)3 · 0.16 = 0.393
µ4 =3∑
i=1[xi − E (X )]4p(xi ) = (1− 1.56)4 · 0.6+
+(2− 1.56)4 · 0.24 + (3− 1.56)4 · 0.16 = 0.756
Jirı Neubauer Random Variables
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Example
First of all we have to calculate 3rd and 4th central moment. Themean of the given random variable is E (X ) = 1.56, the standarddeviation is σ(X ) = 0.753.
µ3 =3∑
i=1[xi − E (X )]3p(xi ) = (1− 1.56)3 · 0.6+
+(2− 1.56)3 · 0.24 + (3− 1.56)3 · 0.16 = 0.393
µ4 =3∑
i=1[xi − E (X )]4p(xi ) = (1− 1.56)4 · 0.6+
+(2− 1.56)4 · 0.24 + (3− 1.56)4 · 0.16 = 0.756
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Example
First of all we have to calculate 3rd and 4th central moment. Themean of the given random variable is E (X ) = 1.56, the standarddeviation is σ(X ) = 0.753.
µ3 =3∑
i=1[xi − E (X )]3p(xi ) = (1− 1.56)3 · 0.6+
+(2− 1.56)3 · 0.24 + (3− 1.56)3 · 0.16 = 0.393
µ4 =3∑
i=1[xi − E (X )]4p(xi ) = (1− 1.56)4 · 0.6+
+(2− 1.56)4 · 0.24 + (3− 1.56)4 · 0.16 = 0.756
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Example
The skewness is equal to
α3 =µ3
σ3= 0.922,
the kurtosis isα4 =
µ4
σ4− 3 = −0.644.
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Example
The skewness is equal to
α3 =µ3
σ3= 0.922,
the kurtosis isα4 =
µ4
σ4− 3 = −0.644.
Jirı Neubauer Random Variables
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Example
Calculate the skewness and the kurtosis of the random X with theprobability density function
f (x) =
12x2(1− x) 0 < x < 1,
0 otherwise.
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Example
The mean of the given random variable is E (X ) = 3/5. thestandard deviation is 1/5.
First of all we calculate 3rd and 4th
central moment.
µ3 =
1∫0
[x − 0.6]312x2(1− x)dx = · · · = − 2
875= −0.00229,
µ4 =
1∫0
[x − 0.6]412x2(1− x)dx = · · · = 33
8750= 0.00377.
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Example
The mean of the given random variable is E (X ) = 3/5. thestandard deviation is 1/5. First of all we calculate 3rd and 4th
central moment.
µ3 =
1∫0
[x − 0.6]312x2(1− x)dx = · · · = − 2
875= −0.00229,
µ4 =
1∫0
[x − 0.6]412x2(1− x)dx = · · · = 33
8750= 0.00377.
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Example
The mean of the given random variable is E (X ) = 3/5. thestandard deviation is 1/5. First of all we calculate 3rd and 4th
central moment.
µ3 =
1∫0
[x − 0.6]312x2(1− x)dx = · · · = − 2
875= −0.00229,
µ4 =
1∫0
[x − 0.6]412x2(1− x)dx = · · · = 33
8750= 0.00377.
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Example
The skewness is equal to
α3 =µ3
σ3= −2
7= −0.286,
the kurtosis is
α4 =µ4
σ4− 3 = − 9
14= −0.643.
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Example
The skewness is equal to
α3 =µ3
σ3= −2
7= −0.286,
the kurtosis is
α4 =µ4
σ4− 3 = − 9
14= −0.643.
Jirı Neubauer Random Variables