Introduction to Random Variables

Introduction to Random Variables
1 Definition of random variable
Sometimes, it is not enough to describe all possible results of an
experiment:
1 Definition of random variable
2 Discrete and continuous random variable
Toss a coin 3 times: {(HHH), (HHT), …}
Throw a dice twice: {(1,1), (1,2), (1,3), …}
Probability function
Distribution function
Density function
Some tine it is useful to associate a number to each result of an experiment
Define a variable
3 Characteristic measures of a random variable
We don’t know the result of the experiment before we carry it out
We don’t know the value of the variable before the experiment
Mean, variance
Other measures
4 Transformation of random variables
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1 Definition of random variable
1 Definition of random variable
A random variable is a function which associates a
real number to each element of the sample space
Sometimes, it is not enough to describe all possible results of an
experiment:
Toss a coin 3 times: {(HHH), (HHT), …}
Throw a dice twice: {(1,1), (1,2), (1,3), …}
Random Variables are represented in capital letters, generally
the last letters of the alphabet: X,Y, Z, etc.
A veces es útil asociar un número a cada resultado del experimento.
X = Number of head on the first toss X[(HHH)]=1, X[(THT)]=0, …
No conocemos el resultado del experimento antes de realizarlo
Y = Sum of points
Y[(1,1)]=2, Y[(1,2)]=3, …
The values taken by the variable are represented by small letters,
No conocemos el valor que va a tomar la variable antes del experimento
x=1 is a possible value of X
y=3.2 is a possible value of Y
z=-7.3 is a possible value of Z
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1 Definition of random variable
1 Definition of random variable
Examples
E
si
X(sk) = a
Number of defective units in a random sample of 5 units
Number of faults per
cm2 of
X(si) = b; si ∈ E
sk
RX
material
a
b
Lifetime of a lamp
Resistance to compression of concrete
• The space RX is the set of ALL possible values of X(s).
• Each possible event of E has an associated value in RX
• We can consider Rx as another random space
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Introduction to Random Variables
1 Definition of random variable
E
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si
X(si) = b; si ∈ E
sk
1 Definition of random variable
X(sk) = a
22 Discrete
Discrete and
and continuos
continuousrandom
randomvariables
variable
RX
a
Probability function
Distribution function
Density function
b
The elements in E have a probability distribution, this distribution is also
associated to the values of the variable X. That is, all r.v. preserve the
probability structure of the random experiment that generates it:
3 Characteristic measures of a random variable
Mean, variance
Other measures
Pr( X = x) = Pr( s ∈ E : X ( s ) = x)
4 Transformation of random variables
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2 Discrete and continuous random variables
2 Discrete and continuous random variables
Examples
Examplesof
ofdiscrete
discreterandom
randomvariables
variables
The rank of a random variable una variable aleatoria is the set of
possible values taken by the variable.
Number
Numberof
offaults
faultson
onaaglass
glasssurface
surface
Depending on the rank, the variables can be classified as:
Proportion
Proportionof
ofdefault
defaultparts
partsininaasample
sampleof
of1000
1000
Generally count
the number of times
that something
happens
Number
Numberof
ofbits
bitstransmited
transmitedand
andreceived
receivedcorrectly
correctly
Discrete:
Discrete:Those
Thosethat
thattake
takeaafinite
finiteor
orinfinite
infinite(numerable)
(numerable)number
numberof
ofvalues
values
Examples
Examplesof
ofcontinuous
continuous random
randomvariables
variables
Continuous:
Continuous:Those
Thosewhose
whoserank
rankisisan
aninterval
intervalof
ofreal
realnumbers
numbers
Electric
Electriccurrent
current
Longitude
Longitude
Generally measure a
magnitude
Temperature
Temperature
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Weight
Weight
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2 Discrete random variables
2 Discrete random variables
The values taken by a random variable change from one experiment
to another, since the results of the experiment are different
The properties of the probability function come from the axioms of
probability:
A r.v. is defined by
1.
3.
The values that it takes.
The probability of taking each value.
0≤P(A) ≤1 2. P(E)=1
P(AUB)=P(A)+P(B) si A∩B=Ø
0 ≤ p ( xi ) ≤ 1
p(xi)
n
∑ p( x ) = 1
i =1
i
a < b < c → A = {a ≤ X ≤ b} B = {b < X ≤ c}
Pr(a ≤ X ≤ c) = Pr(a ≤ X ≤ b) + Pr(b < X ≤ c)
This is a function that indicates the probability of each
possible value
x
p ( xi ) = P ( X = xi )
x1
x2
x3
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x4
x5
x6
xn
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2 Discrete random variables
2 Discrete random variables
Experiment: Toss 2 coins.
X=Number of tails.
Experiment: Toss 2 coins.
X=Number of tails.
0
E
HH
TH
HT
TT
1/
4
1/
2
Pr
H
H
X
P(X=x)
H
T
0
1/4
1
1/2
2
1/4
1
X
RX
0
1
T
H
T
T
2
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2 Discrete random variables
2 Discrete random variables
Experiment: Toss 2 coins.
X=Number of tails.
p(x)
x=0
x=1
x=2
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Sometimes we might be interested on the probability that a variable
takes a value less or equal to a quantity
X
P(X=x)
0
1/4
1
1/2
F (−∞) = 0 F (+∞) = 1
if X takes values x1 ≤ x 2 ≤ K ≤ x n :
2
1/4
F ( x1 ) = P( X ≤ x1 ) = p ( x1 )
F ( x0 ) = P ( X ≤ x0 )
F ( x2 ) = P( X ≤ x2 ) = p ( x1 ) + p ( x2 )
M
X
F ( xn ) = P( X ≤ xn ) = ∑ i =1 p( xi ) = 1
n
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2 Discrete random variables
2 Discrete random variables
Experiment: Toss 2 coins.
X=Number of tails.
Experiment: Toss 2 coins.
X=Number of tails.
p(x)
x=0
x=1
x=2
X
P(X=x)
0
1/4
1
1/2
2
1/4
F(x)
X
1
0.75
0.5
0.25
X
x=0
x=1
x=2
F(x)
0
1/4
1
3/4
2
1
X
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2 Continuous random variables
2 Continuous random variables
When a random variable is continuous, it doesn’t make sense to sum:
Density function describes the probability distribution of a continuous
random variable. It is a function that satisfies:
∞
∑ p( x ) = 1
i =1
i
Since the set of of values taken by the variable is not numerable
We can generalize
f ( x) ≥ 0
∑→ ∫
∫
+∞
−∞
We introduce a new concept instead of the probability function of
discrete random variables
f ( x) dx = 1
b
P(a ≤ X ≤ b) = ∫ f ( x) dx
a
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2 Continuous random variables
2 Continuous random variables
Density function describes the probability distribution of a continuous
random variable. It is a function that satisfies:
a
P( X = a) = ∫ f ( x) dx = 0
a
f ( x) ≥ 0
∫
+∞
−∞
P ( a ≤ X ≤ b) = P ( a < X ≤ b)
= P ( a ≤ X < b)
f ( x) dx = 1
b
P(a ≤ X ≤ b) = ∫ f ( x) dx
a
a
= P ( a < X < b)
b
a
Area below the curve
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2 Continuous random variables
2 Continuous random variables
0.5
If we measure a continuous variable and represent the values in a
histogram:
0.4
The density function
doesn’t have to be
symmetric, or be
define for all values
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the form of the
curve will
depend on one
or more
parameters
Estadística, Profesora: María Durbán
0.0
0.1
0.2
x2
0.3
fX (x | β )
0
5
10
15
20
25
30
If we make the intervals smaller and smaller:
y
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2 Continuous random variables
2 Continuous random variables
f ( x)
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2 Continuous random variables
2 Continuous random variables
Example
„
Example
The density function of the use of a machine in a year
(in hours x100):
„
What is the probability that a machine randomly selected has
been used less than 320 hours?
f(x)
⎧ 0.4
⎪ 2.5 x,
⎪⎪
0.4
f ( x ) = ⎨0.8 −
x,
2.5
⎪
⎪0,
⎩⎪
f(x)
P ( X < 3. 2 ) =
0.4
0 < x < 2.5
2. 5
3. 2
0. 4 ⎞
⎛
⎛ 0. 4 ⎞
x ⎟dx + ∫ ⎜ 0.8 −
x ⎟dx
= ∫⎜
2. 5 ⎠
2. 5 ⎠
0⎝
2.5⎝
2.5 ≤ x < 5
else
elsewhere
2.5
5
x
= 0.74
2.5
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0.4
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3.2
5
x
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2 Continuous random variables
2 Continuous random variables
As in the case of discrete random variables, we can define the distribution
of a continuous random variables by means of the Distribution function:
F ( x) = P( X ≤ x) = ∫
x
−∞
f (u ) du
As in the case of discrete random variables, we can define the distribution
of a continuous random variables by means of the Distribution function:
F ( x) = P( X ≤ x) = ∫
−∞ < x < ∞
x
−∞
−∞ < x < ∞
f (u ) du
In the discrete case, the Probability function is obtained as the
difference of to adjoin values of F(x). In the case of continuous
variables:
P( X ≤ x)
f ( x) =
x
dF ( x)
dx
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2 Continuous random variables
2 Continuous random variables
The Distribution function satisfies the following properties:
The Distribution function satisfies the following properties:
a < b ⇒ F (a ) ≤ F (b) It is non-decreasing
F (−∞) = 0 F (+∞) = 1 It is right-continuous
a < b ⇒ F (a ) ≤ F (b)
F (−∞) = 0 F (+∞) = 1
If we define the following disjoint events:
−∞
F (+∞) = Pr( X ≤ +∞) = ∫
+∞
−∞
{ X ≤ a} {a < X ≤ b} → { X ≤ a} ∪ {a < X ≤ b} = { X ≤ b}
Pr( X ≤ b) = Pr( X ≤ a ) + Pr(a < X ≤ b) ≤ F (b)
F (−∞) = Pr( X ≤ −∞) = ∫
First axiom of
probability
−∞
f ( x)dx = 0
f ( x)dx = 1
≥0
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Third axiom of probability
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2 Continuous random variables
2 Continuous random variables
Example
Example
„
⎧ 0.4
⎪ 2.5 x, 0 < x < 2.5
⎪
0.4
⎪
f ( x) = ⎨0.8 −
x, 2.5 ≤ x < 5
2.5
⎪
⎪0, elsewhere
⎪
⎩
The density function of the use of a machine in a year
(en horas x100):
f(x)
0.4
⎧ 0.4
0 < x < 2.5
⎪ 2.5 x,
⎪⎪
0.4
f ( x ) = ⎨0.8 −
x, 2.5 ≤ x < 5
2.5
⎪
else
elsewhere
⎪0,
⎪⎩
Pr(0 < X < 2.5)
2.5
5
x
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Pr(2.5 ≤ X < x)
⎧ x 0.4
0 < x < 2.5
u du
⎪ ∫0
⎪ 2.5
x
0.4
⎪ 2.5 0.4
u du, 2.5 ≤ x < 5
F ( x) = ⎨ ∫
u du + ∫ 0.8 −
0 2.5
2.5
2.5
⎪
⎪
⎪
x≥5
Pr( X ≤ 5)
⎩1
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2 Continuous random variables
2 Continuous random variables
Example
Example
Example
P(x<3.2)
P(x<3.2)
x=3.2
⎧
⎪0.08 x 2
0 < x < 2.5
⎪⎪
2
F ( x) = ⎨-1 + 0.8 x - 0.08 x 2.5 ≤ x < 5
⎪
⎪
x≥5
⎪⎩1
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Introduction to Random Variables
3 Characteristic measures of a r.v.
Central measures
1 Definition of random variable
In the case of a sample of data, the sample mean allocates a weight of
1/n to each value:
1
1
1
x = x1 + x2 + K + xn
n
n
n
2 Discrete and continuous random variable
Probability function
Distribution function
Density function
The mean μ or Expectation of a r.v. uses the probability as a weight:
33Characteristic
Characteristicmeasures
measuresofofaarandom
randomvariable
variable
μ = E [ X ] = ∑ xi p( xi )
Mean, variance
Other measures
discrete r.v.
i
+∞
μ = E [ X ] = ∫ x f ( x) dx
−∞
4 Transformation of random variables
continuous r.v.
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3 Characteristic measures of a r.v.
3 Characteristic measures of a r.v.
Central measures
Example
„
⎧ 0.4
⎪ 2.5 x, 0 < x < 2.5
⎪
0.4
⎪
f ( x) = ⎨0.8 −
x, 2.5 ≤ x < 5
2.5
⎪
⎪0, elsewhere
⎪
⎩
Intuitively: Median = value that divides the total probability in to parts
P( X ≤ m) = 0.5
F (m) ≥ 0.5
0.5
E[X ] = ∫
0.5
+∞
−∞
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What is the average time of use of the machines?
xf ( x)dx = ∫
= 2.5
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2.5
0
5
0.4 2
0.4 2
x dx + ∫ 0.8 x −
x dx
2.5
2.5
2.5
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3 Characteristic measures of a r.v.
3 Characteristic measures of a r.v.
Other measures
Example
„
If we want to know the time of use such that 50% of the machines
have a use less or equal to that value
The percentil p of a random variable is the value xp that satisfies:
F (m) = 0.5
⎧0.08 x 2
⎪
F ( x) = ⎨-1 + 0.8 x - 0.08 x 2
⎪1
⎩
0 < x < 2.5
2.5 ≤ x < 5
x≥5
p( X < x p ) ≤ p y p( X ≤ x p ) ≥ p
discrete r.v.
F (xp ) = p
continuous r.v.
A special case are quartiles which divide the distribution in 4 parts
0.08 x 2 = 0.5 → m = 2.5
Q1 = p0.25
-1 + 0.8 x - 0.08 x 2 = 0.5 → m = 2.5
Q2 = p0.5 = Median
Q3 = p0.75
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3 Characteristic measures of a r.v.
3 Characteristic measures of a r.v.
Medisures of dispersion
Medisures of dispersion
2
Var [ X ] = E ⎡( X − E [ X ]) ⎤
⎣
⎦
2
Var [ X ] = E ⎡( X − E [ X ]) ⎤
⎣
⎦
The sample variance of a set of data is given by:
Var [ X ] = E ⎡⎣ X 2 ⎤⎦ − ( E [ X ])
1
1
1
s2 = (x1 −x)2 + (x2 −x)2 +K+ (xn −x)2
n
n
n
The Variance of a r.v. also uses the probability as a weight:
σ = Var [ X ] = ∑ ( xi − μ ) p( xi )
2
2
2
2
E ⎡( X − E [ X ]) ⎤ = E ⎡ X 2 + ( E [ X ]) − 2 XE [ X ]⎤
⎣
⎦
⎣
⎦
= E ⎡⎣ X 2 ⎤⎦ + ( E [ X ]) − 2 E [ X ] E [ X ]
2
discrete r.v.
= E ⎡⎣ X 2 ⎤⎦ − ( E [ X ])
i
+∞
σ 2 = Var [ X ] = ∫ ( x − μ ) 2 f ( x) dx
Estadística, Profesora: María Durbán −∞
continuous r.v.
2
43
2
E [ X ] is a constant,
does not depend on X
It is a linear operator
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Introduction to Random Variables
4 Transformation of random variables
In some situations we will need to know the probability distribution of a
transformation of a random variable
1 Definition of random variable
2 Discrete and continuous random variable
Examples
Probability function
Distribution function
Density function
Change units
Use logarithmic scale
aX + b
sinXX
sin
3 Characteristic measures of a random variable
11
XX
Mean, variance
Other measures
4 Transformation
random
variables
4 Transformation
of of
random
variables
Y = g( X )
e
45
|X |
X
log X
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X2
X
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4 Transformation of random variables
4 Transformation of random variables
Example
Let X be a r.v. If we change to Y=h(X), we obtain a new r.v.:
A company packs microchips in lots. It is know that the probability
distribution of the number of microchips per lots is given by:
Distribution function
Y = h( X )
FY ( y ) = Pr(Y ≤ y ) = Pr(h( X ) ≤ y ) = Pr( x ∈ A)
A = { x, h( x) ≤ y}
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x
p(x)
F(x)
11
0.03
0.03
12
0.03
0.06
13
0.03
0.09
14
0.06
0.15
15
0.26
0.41
16
0.09
0.5
17
0.12
0.62
18
0.21
0.83
19
0.14
0.97
20
0.03
1
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¿ Pr( X 2 ≤ 144)?
Pr( X 2 ≤ 144) = Pr( x ∈ A)
{
A = x, x ≤ 144
}
A = { x, x 2 ≤ 144}
Pr ( X ≤ 12 ) = 0.06
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4 Transformation of random variables
4 Transformation of random variables
Density function
Y = h( X )
If X is a continuous r.v. Y=h(X), where h is derivable and inyective
In general:
If
h
is continuous and monotonic increasing :
fY ( y ) = f X ( x )
FY ( y ) = Pr(h( X ) ≤ y ) = Pr( X ≤ h −1 ( y )) = FX (h −1 ( y ))
If
x
⎧ ∂FX ( x) dx
dx dy
∂FY ( y ) ∂Fx (h ( y )) ⎪⎪
fY ( y ) =
=
=⎨
∂y
∂y
⎪ ∂ (1 − FX ( x)) dx
⎪⎩
dx
dy
h is continuous and monotonic decreasing:
−1
dx
dy
increasing
−1
−1
FY ( y ) = Pr(h( X ) ≤ y ) = Pr( X ≥ h ( y )) = 1 − FX (h ( y ))
decreasing
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4 Transformation of random variables
4 Transformation of random variables
Example
If X is a continuous r.v. Y=h(X), where h is derivable and inyective
fY ( y ) = f X ( x )
The velocity of a gas particle is a r.v. V with density function
dx
dy
(b 2 / 2)v 2 e − bv v > 0
fV (v) =
0
elsewhere
For discrete r.v.:
The kinetic energy of the particle is
function of W?
pY ( y ) = Pr(Y = y ) =
∑
h ( xi ) = y
What is the density
Pr( X = xi )
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W = mV 2 / 2
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4 Transformation of random variables
4 Transformation of random variables
Example
Example
The velocity of a gas particle is a r.v. V with density function
The velocity of a gas particle is a r.v. V with density function
(b 2 / 2)v 2 e − bv v > 0
fV (v) =
0
elsewhere
(b 2 / 2)v 2 e − bv v > 0
fV (v) =
0
elsewhere
W = mV 2 / 2 → v = 2w / m v = − 2w / m
dv
1
=
dw
2mw
fV (h −1 ( w)) = (b 2 / 2)
(
)e
2
2w / m
(b 2 / 2m) 2w / m e − b 2 w / m w > 0
fW ( w) =
0
elsewhere
−b 2 w / m
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4 Transformation of random variables
4 Transformation of random variables
Expectation
+∞
E [ h( X ) ] =
Y = h( X )
∫
∑
−∞
Expectation
+∞
h( x) f X ( x)dx
xi , h ( xi ) = y
E [ h( X ) ] =
h( xi ) p ( X = xi )
∫
∑
−∞
h( x) f X ( x)dx
xi , h ( xi ) = y
increasing
h( xi ) p ( X = xi )
Linear Transformations
E [ y] = ∫
+∞
−∞
yf y ( y )dy = ∫
+∞
−∞
dx
h( x) f X ( x) dy
dy
Y = a + bX
E [Y ] = a + bE [ X ]
Var [Y ] = b 2Var [ X ]
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