Introduction to Random Variables 1 Definition of random variable · 2009-02-10 · Introduction to...

Introduction to Random VariablesIntroduction to Random Variables

1 Definition of random variable1 Definition of random variable

2 Discrete and continuous random variable

P b bilit f tiProbability function Distribution functionDensity function

3 Characteristic measures of a random variable

Mean, varianceOther measures

Estadística, Profesora: María Durbán1

4 Transformation of random variables


Sometimes, it is not enough to describe all possible results of an experiment:p

Toss a coin 3 times: {(HHH), (HHT), …}Throw a dice twice: {(1 1) (1 2) (1 3) }Throw a dice twice: {(1,1), (1,2), (1,3), …}

Some tine it is useful to associate a number to each result of an experiment

Define a variable

We don’t know the result of the experiment before we carry it outWe 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



Sometimes, it is not enough to describe all possible results of an experiment:p

Toss a coin 3 times: {(HHH), (HHT), …}Throw a dice twice: {(1 1) (1 2) (1 3) }

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, …

Throw a dice twice: {(1,1), (1,2), (1,3), …}

No conocemos el resultado del experimento antes de realizarlo

No conocemos el valor que va a tomar la variable antes del experimento

Y = Sum of points Y[(1,1)]=2, Y[(1,2)]=3, …

No conocemos el valor que va a tomar la variable antes del experimento



A random variable is a function which associates a real number to each element of the sample space

R d V i bl t d i it l l tt llRandom Variables are represented in capital letters, generallythe last letters of the alphabet: X,Y, Z, etc.

The values taken by the variable are represented by small letters,

x=1 is a possible value of X y=3.2 is a possible value of Yz=-7.3 is a possible value of Z


z 7.3 is a possible value of Z


ExamplesExamples

Number of defective units in a random sample of 5 units

Number of faults per cm2 of materialNumber of faults per cm2 of material

Lifetime of a lamp

Resistance to compression of concrete


1 Definition of random variable

si


E X(si) = b; si ∈ Esk

RX

X(sk) = a

a b

• The space RX is the set of ALL possible values of X(s).

• Each possible event of E has an associated value in R• Each possible event of E has an associated value in RX

• We can consider Rx as another random space



si


E X(si) = b; si ∈ Esk

RX

X(sk) = a

a b

Th l t i E h b bilit di t ib ti thi di t ib ti i lThe elements in E have a probability distribution, this distribution is alsoassociated to the values of the variable X. That is, all r.v. preserve the probability structure of the random experiment that generates it:

Pr( ) Pr( : ( ) )X x s E X s x= = ∈ =




2 Discrete and continuos random variables

P b bilit f ti


Probability function Distribution functionDensity function





2 Discrete and continuous random variables2 Discrete and continuous random variables

The rank of a random variable is the set of possible values taken by the variable.

Depending on the rank, the variables can be classified as:

Discrete: Those that take a finite or infinite (numerable) number of values

Continuous: Those whose rank is an interval of real numbers

Discrete: Those that take a finite or infinite (numerable) number of values

Continuous: Those whose rank is an interval of real numbersContinuous: Those whose rank is an interval of real numbersContinuous: Those whose rank is an interval of real numbers


2 Discrete and continuous random variables

Examples of discrete random variablesExamples of discrete random variables

2 Discrete and continuous random variables

Examples of discrete random variables

Number of faults on a glass surface

Examples of discrete random variables

Number of faults on a glass surface

Proportion of default parts in a sample of 1000

N b f bit t itt d d i d tl

Proportion of default parts in a sample of 1000

N b f bit t itt d d i d tlGenerally count

Number of bits transmitted and received correctly

Examples of continuous random variables

Number of bits transmitted and received correctly

Examples of continuous random variables

the number of times that somethinghappens

Electric currentElectric current

Longitude

Temperature

Longitude

Temperature

Generally measure a magnitude


Temperature

Weight

Temperature

Weight

2 Discrete random variables2 Discrete random variables

The values taken by a random variable change from one experimentThe values taken by a random variable change from one experiment to another, since the results of the experiment are different

A r.v. is defined by y

The values that it takes.Th b bilit f t ki h lThe probability of taking each value.

Thi i f ti th t i di t th b bilit f hThis is a function that indicates the probability of each possible value

( ) ( )i ip x P X x= =


( ) ( )i ip


The properties of the probability function come from the axioms of

probability:

p(x ) 0 ( ) 1ip x≤ ≤

1. 0≤P(A) ≤1 2. P(E)=1 3. P(AUB)=P(A)+P(B) si A∩B=Ø

p(xi)

{ } { }1

( ) 1n

ii

p x

a b c A a X b B b X c=

=

< < → = ≤ ≤ = < ≤

∑{ } { }

Pr( ) Pr( ) Pr( )a b c A a X b B b X c

a X c a X b b X c< < → = ≤ ≤ = < ≤

≤ ≤ = ≤ ≤ + < ≤

x

x1 x 2 x3 x4 x5 x6 xn


2 Discrete random variables

Experiment: Toss 2 coins


Experiment: Toss 2 coins. X=Number of tails.

HH TH TT

E

HH THHT TT

Pr

RX

X

0 1 2






X P(X=x)H H ( )

0 1/4TH1 1/2

2 1/4T H 2 1/4 T

T T

H


T T





X P(X=x)p(x) ( )

0 1/4

1 1/2

2 1/42 1/4

x=0 x=1 x=2 X


x 0 x 1 x 2 X


Sometimes we might be interested on the probability that a variable takes a value less or equal to a quantity

0 0( ) ( )F x P X x= ≤0 0

1 2 n

( ) ( )( ) 0 ( ) 1

if X takes values x x : ( ) ( ) ( )

F Fx

F x P X x p x

−∞ = +∞ =≤ ≤ ≤

≤K

1 1 1

2 2 1 2

( ) ( ) ( )( ) ( ) ( ) ( )

F x P X x p xF x P X x p x p x

= ≤ == ≤ = +

M


1( ) ( ) ( ) 1n

n n iiF x P X x p x

== ≤ = =∑





X P(X=x)p(x) ( )

0 1/4

1 1/2

2 1/42 1/4

x=0 x=1 x=2 X


x 0 x 1 x 2 X





X F(x)F(x) ( )

0 1/4 0 75

1

1 3/4

2 10 25

0.5

0.75

2 1

x=0 x=1 x=2 X

0.25


x 0 x 1 x 2 X

2 Continuous random variables2 Continuous random variables

When a random variable is continuous it doesn’t make sense to sum:When a random variable is continuous, it doesn t make sense to sum:

( ) 1ip x∞

=∑1i=∑

Since the set of values taken by the variable is not numerable

We can generalize →∑ ∫

We introduce a new concept instead of the probability function of discrete random variablesdiscrete random variables



Density function describes the probability distribution of a continuousrandom variable. It is a function that satisfies:

( ) 0f x ≥( ) 0

( ) 1

f x

f x dx+∞

≥

=∫ ( ) 1

( ) ( )b

f x dx

P a X b f x dx

−∞

≤ ≤ =

∫∫( ) ( )

aP a X b f x dx≤ ≤ = ∫



Density function describes the probability distribution of a continuousrandom variable. It is a function that satisfies:

( ) 0f x ≥( ) 0

( ) 1

f x

f x dx+∞

≥

=∫ ( ) 1

( ) ( )b

f x dx

P a X b f x dx

−∞

≤ ≤ =

∫∫ a b( ) ( )

aP a X b f x dx≤ ≤ = ∫

Area below the curve



( ) ( ) 0a

aP X a f x dx= = =∫

( ) ( )( )

P a X b P a X bP X b

≤ ≤ = < ≤≤ ( )

( )P a X bP a X b

= ≤ <= < < a



0.5

The density function d ’t h t b

30.

4doesn’t have to be symmetric, or be define for all values ( | )Xf x β

x2

0.2

0.3

0.1

the form of the

0 5 10 15 20 25 30

0.0

the form of the curve will depend on one


y

0 5 10 15 20 25 30or more parameters


If we measure a continuous variable and represent the values in a histogram:


If we make the intervals smaller and smaller:




( )f x


2 Continuous random variables

Example


The density function of the use of a machine in a year (i h 100)

Example

(in hours x100):

f(x)

⎪⎪⎧ << 2.5x0x,

2 50.4

0.4

⎪

⎪⎪⎪

⎨ <≤−=

l

5x2.5x,2.50.40.8

2.5)x(f

⎪⎪⎪

⎩else0, elsewhere


2.5 5x


Example


What is the probability that a machine randomly selected has b d l th 320 h ?

Example

been used less than 320 hours?

f(x)

0.423 ).( =<XP

524080

524052

0

23

52 ...

.

.. .

.

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛= ∫ ∫ dxxdxx

740.=


2.5 5x

3.2


As in the case of discrete random variables, we can define the distribution of a continuous random variables by means of the Distribution function:

( ) ( ) ( ) x

F x P X x f u du x= ≤ = −∞ < < ∞∫( ) ( ) ( )f−∞∫

( )P X x≤

29x


As in the case of discrete random variables, we can define the distribution of a continuous random variables by means of the Distribution function:

( ) ( ) ( ) x

F x P X x f u du x= ≤ = −∞ < < ∞∫( ) ( ) ( )f−∞∫

In the discrete case, the Probability function is obtained as the

( )dF

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:

( )( ) dF xf xdx

=



The Distribution function satisfies the following properties:

( ) ( )a b F a F b< ⇒ ≤ It is non-decreasing( ) ( )( ) 0 ( ) 1F F−∞ = +∞ =

gIt is right-continuous

If we define the following disjoint events:

{ } { } { } { } { }X a a X b X a a X b X b≤ < ≤ → ≤ ∪ < ≤ = ≤{ } { } { } { } { } X a a X b X a a X b X b≤ < ≤ → ≤ ∪ < ≤ = ≤

Pr( ) Pr( ) Pr( ) ( )X b X a a X b F b≤ = ≤ + < ≤ ≤

First axiom of probability

Estadística, Profesora: María Durbán31Third axiom of probability

0≥


The Distribution function satisfies the following properties:

( ) ( )a b F a F b< ⇒ ≤( ) ( )( ) 0 ( ) 1F F−∞ = +∞ =

( ) Pr( ) ( ) 0F X f x dx−∞

−∞−∞ = ≤ −∞ = =∫

( ) Pr( ) ( ) 1F X f x dx+∞

−∞+∞ = ≤ +∞ = =∫



Example


The density function of the use of a machine in a year ( h 100)

Example

(en horas x100):

f(x)

⎪⎪⎧ << 2.5x0x,

2 50.4

0.4

⎪

⎪⎪⎪

⎨ <≤−=

l

5x2.5x,2.50.40.8

2.5)x(f

⎪⎪⎪

⎩else0, elsewhere


2.5 5x


Example


0.4 x, 0 x 2.52 5

⎧ < <⎪2.50.4( ) 0.8 x, 2.5 x 52.5

f x

⎪⎪⎪= − ≤ <⎨⎪ 2.5

0, elsewhere⎪⎪⎪⎩

Pr(0 2 5)X< < Pr(2 5 )X x≤ <

0

0.4 0 x 2.52.5

xu du

⎧< <⎪

⎪∫

Pr(0 2.5)X< < Pr(2.5 )X x≤ <

2.5

0 2.5

0.4 0.4( ) 0.8 u du, 2.5 x 52.5 2.5

xF x u du

⎪⎪= + − ≤ <⎨⎪⎪

∫ ∫

Estadística, Profesora: María Durbán341 x 5

⎪⎪ ≥⎩ Pr( 5)X ≤


ExampleExample


Example

P(x<3.2)

x=3 2x=3.2

20 08 0 2 5x x⎧⎪ < <⎪

2

0.08 0 2.5( ) -1 0.8 - 0.08 2.5 5

x xF x x x x

< <⎪⎪= + ≤ <⎨⎪⎪


1 5x⎪

≥⎪⎩


Example


Example

P(x<3.2)






3 Characteristic measures of a random variable3 Characteristic measures of a random variable




3 Characteristic measures of a r v3 Characteristic measures of a r.v.

Central measures

In the case of a sample of data, the sample mean allocates a weight of 1/n to each value:

1 1 1

Th E t ti f th b bilit i ht

1 21 1 1

nx x x xn n n

= + + +K

The mean or Expectation of a r.v. uses the probability as a weight:μ

[ ] ( )i iE X x p xμ = =∑ discrete r.v.[ ]

[ ]

( )

( )

i ii

x p x

E X x f x dx

μ

μ+∞

= =

∑

∫

discrete r.v.

continuous r.v.


[ ] ( )fμ−∞∫


Central measures

Intuitively: Median = value that divides the total probability in to parts

( ) 0 5P X m≤ =( ) 0.5

( ) 0.5

P X m

F m

≤ =

≥

0.50.5

( ) 0.5F m ≥


3 Characteristic measures of a r v

Example

3 Characteristic measures of a r.v.

What is the average time of use of the machines?

Example

0.4 x, 0 x 2.52.5

⎧ < <⎪⎪

0.4( ) 0.8 x, 2.5 x 52.5

f x⎪⎪= − ≤ <⎨⎪⎪0, elsewhere⎪⎪⎩

[ ]2.5 52 2

0 2.5

0.4 0.4( ) d 0.8 d2.5 2.5

E X xf x dx x x x x x+∞

−∞= = + −∫ ∫ ∫


.5 .5 2.5=

3 Characteristic measures of a r v

Example

3 Characteristic measures of a r.v.

If we want to know the time of use such that 50% of the machines have a use less or equal to that value

Example

have a use less or equal to that value

( ) 0.5F m =2

2

0.08 0 x 2.5( ) -1 0.8 - 0.08 2.5 x 5

xF x x x

⎧ < <⎪= + ≤ <⎨( )

1 x 5⎨⎪ ≥⎩

20 08 0 5 2 52

2

0.08 0.5 2.5-1 0.8 - 0.08 0.5 2.5

x mx x m= → =

+ = → =



Other measures

The percentil p of a random variable is the value xp that satisfies:

( ) y ( )( )

p p

p

p X x p p X x pF x p

< ≤ ≤ ≥

=discrete r.v.

continuous r v( )p p continuous r.v.

A special case are quartiles which divide the distribution in 4 parts

1 0.25

2 0 5 MedianQ pQ p

== =


2 0.5

3 0.75

MedianQ pQ p=


Measures of dispersionp

⎡ ⎤

The sample variance of a set of data is given by:

[ ] [ ]( )2Var X E X E X⎡ ⎤= −

⎣ ⎦The sample variance of a set of data is given by:

2 2 2 21 2

1 1 1( ) ( ) ( )ns x x x x x xn n n

= − + − + + −K

The Variance of a r.v. also uses the probability as a weight:

n n n

[ ] 22 ( ) ( )i ii

Var X x p xσ μ= = −∑ discrete r.v.

Estadística, Profesora: María Durbán43[ ]2 2( ) ( )

i

Var X x f x dxσ μ+∞

−∞= = −∫ continuous r.v.


Measures of dispersion

⎡ ⎤

p

[ ] [ ]( )2Var X E X E X⎡ ⎤= −

⎣ ⎦

[ ] [ ]( )22Var X E X E X⎡ ⎤= −⎣ ⎦

[ ]( ) [ ]( ) [ ]2 22 2E X E X E X E X XE X⎡ ⎤ ⎡ ⎤− = + −⎣ ⎦ ⎣ ⎦

[ ]( ) [ ] [ ]

[ ]( )

22

22

2E X E X E X E X

E X E X

⎡ ⎤= + −⎣ ⎦

⎡ ⎤= −⎣ ⎦ It i li t

[ ] is a constant, does not depend on E X

X


[ ]( )E X E X⎡ ⎤= ⎣ ⎦ It is a linear operator








4 Transformation of random variables4 Transformation of random variables


In some situations we will need to know the probability distribution of atransformation of a random variable

Examples

Change unitsUse logarithmic scale

bbaX +2Xsin X

1)(XgY = || X

X

1X


XXe

Xlog


L t X b If h t Y h(X) bt iLet X be a r.v. If we change to Y=h(X), we obtain a new r.v.:

( )Y h X

Distribution function

( )Y h X=

( ) Pr( ) Pr( ( ) ) Pr( )YF y Y y h X y x A= ≤ = ≤ = ∈

{ }, ( )A x h x y= ≤



Example


Example

A company packs microchips in lots. It is know that the probabilitydistribution of the number of microchips per lots is given by:

x p(x) F(x)

0 03 0 03 2P ( 144)?X ≤11 0.03 0.03

12 0.03 0.06

13 0.03 0.09

2¿Pr( 144)?X ≤

{ }2 2( ) ( )

{ }, 144A x x= ≤

14 0.06 0.15

15 0.26 0.41

16 0 09 0 5

{ }( )

2 2Pr( 144) Pr( ) , 144

Pr 12 0.06

X x A A x x

X

≤ = ∈ = ≤

≤ =16 0.09 0.5

17 0.12 0.62

18 0.21 0.83


19 0.14 0.97

20 0.03 1


( )Y h X=In general:

If is continuous and monotonic increasing :h1 1( ) Pr( ( ) ) Pr( ( )) ( ( ))Y XF y h X y X h y F h y− −= ≤ = ≤ =

If is continuous and monotonic decreasing:h1 1( ) P ( ( ) ) P ( ( )) 1 ( ( ))F h X X h F h− −≤ ≥ 1 1( ) Pr( ( ) ) Pr( ( )) 1 ( ( ))Y XF y h X y X h y F h y= ≤ = ≥ = −



Density function

If X is a continuous r.v. Y=h(X), where h is derivable and inyective

( ) ( )Y Xdxf y f xdy

=dy

1

( )( ( ))( )

XF x dxdx dyF h yF y −

∂⎧⎪∂∂ ⎪

increasingx

( ( ))( )( )(1 ( ))

xYY

X

dx dyF h yF yf yF x dxy ydx dy

∂∂ ⎪= = = ⎨∂ −∂ ∂ ⎪⎪⎩

decreasing


dx dy⎪⎩g


If X is a continuous r.v. Y=h(X), where h is derivable and inyective

( ) ( )Y Xdxf y f xdy

=dy

For discrete r.v.:

( )( ) Pr( ) Pr( )

i

Y ih x y

p y Y y X x=

= = = =∑


( )ih x y


ExampleExample

The velocity of a gas particle is a r.v. V with density function

2 2( / 2) 0( )

bvb v e vf v

− >=( )

0 elsewhereVf v =

The kinetic energy of the particle is What is the density function of W?

2 / 2W mV=



ExampleExample


2 2( / 2) 0( )

bvb v e vf v

− >=( )

0 elsewhereVf v =

2 / 2 2 / 2 /W mV v w m v w m= → = = −

12

dvdw mw

= ( )21 2 2 /( ( )) ( / 2) 2 / b w m

Vf h w b w m e− −=


2dw mw ( )


ExampleExample


2 2( / 2) 0( )

bvb v e vf v

− >=( )

0 elsewhereVf v =

2 2 /( / 2 ) 2 / 0b w mb m w m e w− >( / 2 ) 2 / 0( ) 0 elsewhereWb m w m e wf w >

=



ExpectationExpectation

+

[ ]( ) ( )

( ) ( ) ( )

X

i i

h x f x dxE h X

h x p X x

+∞

−∞==

∫∑

, ( )( ) ( )

i i

i ix h x y

p=

∑( )Y h X=

increasing

[ ] ( ) ( ) ( )y XdxE y yf y dy h x f x dydy

+∞ +∞

−∞ −∞= =∫ ∫



ExpectationExpectation

+

[ ]( ) ( )

( ) ( ) ( )

X

i i

h x f x dxE h X

h x p X x

+∞

−∞==

∫∑

, ( )( ) ( )

i i

i ix h x y

p=

∑

Linear TransformationsLinear Transformations

Y a bX= +

[ ] [ ]E Y a bE X= +


[ ] [ ]2Var Y b Var X=

Introduction to Random Variables 1 Definition of random variable · 2009-02-10 · Introduction to...

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Transcript of Introduction to Random Variables 1 Definition of random variable · 2009-02-10 · Introduction to...