introduction to random variables 1 definition of random ... · introduction to random variables 1...

Estadística, Profesora: María Durbán1

Introduction to Random Variables

1 Definition of random variable

2 Discrete and continuous random variable

Probability function Distribution functionDensity function

3 Characteristic measures of a random variable

Mean, varianceOther measures

4 Transformation of random variables



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), …}

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 out We don’t know the value of the variable before the experiment



A veces es útil asociar un número a cada resultado del experimento.

No conocemos el resultado del experimento antes de realizarlo

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

X = Number of head on the first toss X[(HHH)]=1, X[(THT)]=0, …

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

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), …}



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

Random 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



Examples

Number of defective units in a random sample of 5 units

Number of faults per cm2 of material

Lifetime of a lamp

Resistance to compression of concrete


a b

RX

E X(si) = b; si ∈ E

X(sk) = a

si

sk

• 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



a b

RX

E X(si) = b; si ∈ E

X(sk) = a

si

sk

The 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

Probability functionDistribution functionDensity function






2 Discrete and continuous random variables

The rank of a random variable una variable aleatoria is the set ofpossible 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 numbers


Examples of discrete random variables

Number of faults on a glass surface

Proportion of default parts in a sample of 1000

Number of bits transmited and received correctly

Examples of continuous random variables

Electric current

Longitude

Temperature

Weight

Examples of discrete random variables

Number of faults on a glass surface

Proportion of default parts in a sample of 1000

Number of bits transmited and received correctly

Examples of continuous random variables

Electric current

Longitude

Temperature

Weight

Generally count the number of times that somethinghappens

Generally measure a magnitude

2 Discrete and continuous random variables


2 Discrete random variables

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

A r.v. is defined by

The values that it takes.The probability of taking each value.

This is a function that indicates the probability of each possible value

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


x

x1 x 2 x3 x4 x5 x6 xn

p(xi)

The properties of the probability function come from the axioms of

probability:

{ } { }1

0 ( ) 1

( ) 1

Pr( ) Pr( ) Pr( )

in

ii

p x

p x

a b c A a X b B b X ca X c a X b b X c

=

≤ ≤

=

< < → = ≤ ≤ = < ≤

≤ ≤ = ≤ ≤ + < ≤

∑

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



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

HH THHT TT

RX

X

0 1 2

0 1/4 1/2 1

Pr

E



X P(X=x)

0 1/4

1 1/2

2 1/4

T

T

T T

H H

H

H




X P(X=x)

0 1/4

1 1/2

2 1/4

x=0 x=1 x=2 X

p(x)




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

0 0

1 2 n

1 1 1

2 2 1 2

1

( ) ( )( ) 0 ( ) 1

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

( ) ( ) ( ) 1nn n ii

F x P X xF F

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

F x P X x p x=

= ≤−∞ = +∞ =

≤ ≤ ≤= ≤ == ≤ = +

= ≤ = =∑

K

M



X P(X=x)

0 1/4

1 1/2

2 1/4

x=0 x=1 x=2 X

p(x)




X F(x)

0 1/4

1 3/4

2 1

x=0 x=1 x=2 X

F(x)

0.25

0.5

0.75

1




2 Continuous random variables

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

1( ) 1i

ip x

∞

=

=∑

Since the set of 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 variables

→∑ ∫


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

( ) 0

( ) 1

( ) ( ) b

a

f x

f x dx

P a X b f x dx

+∞

−∞

≥

=

≤ ≤ =

∫∫



( ) 0

( ) 1

( ) ( ) b

a

f x

f x dx

P a X b f x dx

+∞

−∞

≥

=

≤ ≤ =

∫∫ a b

Area below the curve

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



( ) ( ) 0

( ) ( ) ( ) ( )

a

aP X a f x dx

P a X b P a X bP a X bP a X b

= = =

≤ ≤ = < ≤= ≤ <= < <

∫

a



y

x2

0 5 10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

0.5

The density function doesn’t have to be symmetric, or be define for all values

the form of the curve will depend on one or more parameters

( | )Xf x β



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

If we make the intervals smaller and smaller:





( )f x



The density function of the use of a machine in a year (in hours x100):

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

<≤−

<<

=

else0,

5x2.5x,2.50.40.8

2.5x0x,2.50.4

)x(f

2.5 5

0.4

f(x)

x

Example

elsewhere



What is the probability that a machine randomly selected has been used less than 320 hours?

2.5 5

0.4

f(x)

x

3.2

740

524080

5240

23

52

0

23

52

.

.

....

).(

. .

.

=

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛=

=<

∫ ∫ dxxdxx

XP

Example


29

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−∞

= ≤ = −∞ < < ∞∫

x

( )P X x≤



( ) ( ) ( ) x

F x P X x f u du x−∞

= ≤ = −∞ < < ∞∫

( )( ) dF xf xdx

=

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:

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



The Distribution function satisfies the following properties:

( ) ( )( ) 0 ( ) 1

a b F a F bF F< ⇒ ≤−∞ = +∞ =

If we define the following disjoint events:

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

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

It is non-decreasingIt is right-continuous

Third axiom of probability

0≥

First axiom ofprobability



( ) ( )( ) 0 ( ) 1

a b F a F bF F< ⇒ ≤−∞ = +∞ =

( ) Pr( ) ( ) 0

( ) Pr( ) ( ) 1

F X f x dx

F X f x dx

−∞

−∞

+∞

−∞

−∞ = ≤ −∞ = =

+∞ = ≤ +∞ = =

∫∫

The Distribution function satisfies the following properties:



The density function of the use of a machine in a year (en horas x100):

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

<≤−

<<

=

else0,

5x2.5x,2.50.40.8

2.5x0x,2.50.4

)x(f

2.5 5

0.4

f(x)

x

Example

elsewhere



Example

0

2.5

0 2.5

0.4 0 x 2.52.50.4 0.4( ) 0.8 u du, 2.5 x 52.5 2.5

1 x 5

x

x

u du

F x u du

⎧< <⎪

⎪⎪= + − ≤ <⎨⎪⎪⎪ ≥⎩

∫

∫ ∫

0.4 x, 0 x 2.52.5

0.4( ) 0.8 x, 2.5 x 52.5

0, elsewhere

f x

⎧ < <⎪⎪⎪= − ≤ <⎨⎪⎪⎪⎩

Pr(0 2.5)X< < Pr(2.5 )X x≤ <

Pr( 5)X ≤



ExampleExample

x=3.2

P(x<3.2)

2

2

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

1 5

x xF x x x x

x

⎧⎪ < <⎪⎪= + ≤ <⎨⎪⎪

≥⎪⎩



Example

P(x<3.2)












3 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:

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

1 21 1 1

nx x x xn n n

= + + +K

μ

[ ]

[ ]

( )

( )

i ii

E X x p x

E X x f x dx

μ

μ+∞

−∞

= =

= =

∑

∫

discrete r.v.

continuous r.v.



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

0.50.5

( ) 0.5

( ) 0.5

P X m

F m

≤ =

≥

Central measures


What is the average time of use of the machines?

Example

[ ]2.5 52 2

0 2.5

0.4 0.4( ) d 0.8 d2.5 2.5

2.5

E X xf x dx x x x x x+∞

−∞= = + −

=

∫ ∫ ∫

0.4 x, 0 x 2.52.5

0.4( ) 0.8 x, 2.5 x 52.5

0, elsewhere

f x

⎧ < <⎪⎪⎪= − ≤ <⎨⎪⎪⎪⎩



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

Example

( ) 0.5F m =2

2

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

1 x 5

xF x x x

⎧ < <⎪= + ≤ <⎨⎪ ≥⎩

2

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.

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

1 0.25

2 0.5

3 0.75

MedianQ pQ pQ p

== ==



Medisures of dispersion

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

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

2 2 2 21 2

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

= − + − + + −K

[ ]

[ ]

22

2 2

( ) ( )

( ) ( )

i ii

Var X x p x

Var X x f x dx

σ μ

σ μ+∞

−∞

= = −

= = −

∑

∫

discrete r.v.

continuous r.v.

[ ] [ ]( )2Var X E X E X⎡ ⎤= −

⎣ ⎦



[ ] [ ]( )2Var X E X E X⎡ ⎤= −

⎣ ⎦

[ ] [ ]( )22Var X E X E X⎡ ⎤= −⎣ ⎦

[ ]( ) [ ]( ) [ ]

[ ]( ) [ ] [ ]

[ ]( )

2 22

22

22

2

2

E X E X E X E X XE X

E X E X E X E X

E X E X

⎡ ⎤ ⎡ ⎤− = + −⎣ ⎦ ⎣ ⎦

⎡ ⎤= + −⎣ ⎦

⎡ ⎤= −⎣ ⎦ It is a linear operator

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

X

Medisures of dispersion








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

)(XgY =

baX +2X

|| X

XXe

XlogX

1nis X sin X

1X



Let X be a r.v. If we change to Y=h(X), we obtain a new r.v.:

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

( )Y h X=

{ }, ( )A x h x y= ≤

Distribution function


Example

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

0.03

0.14

0.21

0.12

0.09

0.26

0.06

0.03

0.03

0.03

p(x)

1

0.97

0.83

0.62

0.5

0.41

0.15

0.09

0.06

0.03

F(x)

20

19

18

17

16

15

14

13

12

11

x2¿Pr( 144)?X ≤

{ }( )

2 2Pr( 144) Pr( ) , 144

Pr 12 0.06

X x A A x x

X

≤ = ∈ = ≤

≤ =


{ }, 144A x x= ≤



1 1( ) Pr( ( ) ) Pr( ( )) ( ( ))Y XF y h X y X h y F h y− −= ≤ = ≤ =

( )Y h X=In general:

If is continuous and monotonic increasing :h

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



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

( ) ( )Y Xdxf y f xdy

=

Density function

1

( )( ( ))( )( )

(1 ( ))

X

xYY

X

F x dxdx dyF h yF yf yF x dxy ydx dy

−

∂⎧⎪∂∂ ⎪= = = ⎨∂ −∂ ∂ ⎪⎪⎩

increasing

decreasing

x



For discrete r.v.:

( )( ) Pr( ) Pr( )

i

Y ih x y

p y Y y X x=

= = = =∑

( ) ( )Y Xdxf y f xdy

=

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



Example

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

2 2( / 2) 0( )

0 elsewhere

bv

Vb v e v

f v− >

=

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

2 / 2W mV=



Example

2 2( / 2) 0( )

0 elsewhere

bv

Vb v e v

f 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− −=




Example

2 2( / 2) 0( )

0 elsewhere

bv

Vb v e v

f v− >

=

2 2 /( / 2 ) 2 / 0( ) 0 elsewhere

b w m

Wb m w m e wf w

− >=




Expectation

[ ], ( )

( ) ( )( )

( ) ( )i i

X

i ix h x y

h x f x dxE h X

h x p X x

+∞

−∞

=

==

∫∑

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

+∞ +∞

−∞ −∞= =∫ ∫

( )Y h X=increasing



Expectation

[ ], ( )

( ) ( )( )

( ) ( )i i

X

i ix h x y

h x f x dxE h X

h x p X x

+∞

−∞

=

==

∫∑

Linear Transformations

Y a bX= +

[ ] [ ][ ] [ ]2

E Y a bE X

Var Y b Var X

= +

=

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