Random variables - UC3Mhalweb.uc3m.es/esp/Personal/personas/icascos/esp/2rvariables.pdf · 2. Discrete and continuous random variables Discrete random variables (probability mass

Random variables

Ignacio Cascos Depto. Estadística, Universidad Carlos III 1

2

Outline1. Definition of randomvariable2. Discrete and continuous randomvariables

� Discrete randomvariables (probabilitymassfunctionand cdf)


(probabilitymassfunctionand cdf)� Continuos randomvariables

(density mass function and cdf)

3. Characteristic features of a randomvariable� Location, scatter and shape parameters� Independent randomvariables

Definition of random variable

A random variableassesses a real number to

each possible outcome of the random experiment.


each possible outcome of the random experiment.

It is randombecause we do not know its value

before carrying out the random experiment.

Definition of random variable� Definition. A random variableX is a mapping

X: E → IR, where E is the sample space associated with a random experiment.


e1

e3

e2

X (e3) X (e2) X (e1)

X

IR

Definition of random variable� The eventsare now of the type X∈A,

where A is a subset of IR.Their probabilities are P(X∈A) = P({ e∈E: X(e)∈A}).


Properties:

1. P(X∈A) ≥ 0 ;

2. P(X∈IR) = 1 ;

3. if A1, A2,…⊂IR satisfy Ai∩Aj= ∅ for i ≠ j,

thenP(X∈∪i= 1,∞ Ai)=Σi= 1,∞ P(X∈Ai) .







Discrete and continuous random variablesThe rangeof a random variable is the set of values that

it can assume.

� A random variable is discreteif its range is finite or denumerable.


denumerable.� Examples: number of defective items, number of rolls of a

die until a 5 occurs.

� A random variable is continuousif its range contains an interval.� Example: battery life.

Discrete random variablesGiven a discrete random variable X, its probability mass function assigns to each possible value of the variablethe probability that X assumes such a value.

p: IR → [0,1]


p: IR → [0,1]

x → p(x) = P(X=x)

It satisfies that 0 ≤ p(x) ≤ 1 for every x and if X assumes

n different possible values x1,…,xn , thenΣi p(xi) = 1.

We have P(X∈A) = Σxi∈A p(xi) .

Discrete random variablesLet X be the number of heads

after tossing 3 times a fair

coins. The probability mass

function of X is:

Funcion de probabilidad

0.25

0.30

0.35

Probability mass function


function of X is:

0 1 2 3

numero motores averiados

prob

abili

dad

0.00

0.05

0.10

0.15

0.20

0.25

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

0 0.125

1 0.375

2 0.375

3 0.125

Discrete random variablesThe cumulative distribution function (cdf)of Xat x is the probability that X is smaller or equal to x.

≤


F(x) = P(X ≤ x)1. limx→ −∞ F(x) = 0 ;2. limx→ ∞ F(x) = 1 ;3. F is nondecreasing ;4. F is right-continuous.

Discrete random variablesThe cumulative

distribution function of

a discrete random

0.8

1.0

Funcion de distribucioncdf


a discrete random

variable is stepwise,

F(x) = P(X ≤ x)

= Σxi ≤ x p(xi) -1 0 1 2 3 4

0.0

0.2

0.4

0.6

numero motores averiados

prob

abili

dad

pro

bab

ility

Continuous random variablesSince the range of a continuous random variable is not

denumerable, an expression like Σi p(xi) = 1 makes no sense.

Histogram for the lifetime of 10000 batteries.


Histogram of duracion

duracion

Den

sity

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

Histogram of duracion

duracion

Den

sity

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

Histogram for the lifetime of 10000 batteries.

Continuous random variablesFunction f describes the

curve drawn together

with the histogram on

0.8

1.0


with the histogram on

the right. We have

P(2 ≤ X ≤ 4) ≈ ∫24 f(x)dx

where X is the lifetime of

a battery. 0 2 4 6 8

0.0

0.2

0.4

0.6

dens

idad

den

sity

Continuous random variablesThe density mass functionf describes the probability

distribution of a continuous random variable. It satisfies:

1. f(x) ≥ 0 ;


1. f(x) ≥ 0 ;

2. ∫−∞+∞ f(x)dx = 1 .

3. We haveP(a ≤ X ≤ b) = ∫ab f(x)dx .Given X continuous r.v., it satisfies

� P(X = a) = 0 ;

� P(a ≤ X ≤ b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a < X < b)

Continuous random variablesIn order to compute the cumulative distribution Function (cdf) of a continuous random variable, we must integrate its density mass function,

≤ ∫


F(x) = P(X ≤ x) = ∫−∞x f(t)dt

1. limx→ −∞ F(x) = 0 ;2. limx→ ∞ F(x) = 1 ;3. F is nondecreasing ;4. F is continuous .

Continuous random variablesSince the cumulative distribution

function is a primitive of the

density mass function, deriving

the cdf, we obtain the density

Mean1

Exponential Distribution

cum

ula

tive

pro

bab

ility

0,8

1


the cdf, we obtain the density

mass function,

f(x) = dF(x)/dx .

We are working with

f(x) = e−x if x > 0

F(x) = 1− e−x if x > 0x

cum

ula

tive

pro

bab

ility

-1 0 1 2 3 4 5 6 7 80

0,2

0,4

0,6







Mathematical expectation or meanThe expecationor mean(µ) of a random variable

is a central point with respect to its distribution

� X discrete, µ = E[X] = ∑xip(xi)


� X discrete, µ = E[X] = ∑xip(xi)

� X continuous, µ = E[X] = ∫ xf(x)dx

Properties: Given X,Yand two real numbers, a,b

1. E[a+bX] = a+bE[X] ;

2. E[X+Y] = E[X]+E[Y] .

Mathematical expecation or meanGiven a functiong: IR → IR, the expectation of

randomvariable g(X) can be computed as


� X discrete, E[g(X)] = ∑g(xi)p(xi)

� X continuous, E[g(X)] = ∫ g(x)f(x)dx

MedianThe medianof a random variable X is a value Me

such that

≥ ≥ ≥


F(Me) ≥ 1/2 ; P(X ≥ Me) ≥ 1/2

If X is a continuous r.v., then F(Me) = 1/2.

QuantilesFor 0 < α < 1, the α-quantileof random variable X is a

value xα such that the probability of X being not greater

than xα is, at least, α and the probability of X being not

−α


smaller than xα is, at least, 1−α.

F(xα) = P(X ≤ xα) ≥ α ; P(X ≥ xα) ≥ 1−α Percentilesand quartilesare defined in a similar way

Pa = xa/100 ; Qi = P25i

where1 ≤ a ≤ 99 and 1 ≤ i ≤ 3 .

Scatter parametersThe varianceof a random variable X is given by

σ2 = Var[X] = E[(X−E[X])2]

� X discrete, σ2 = Var[X] = ∑(xi−µ)2 p(xi)


� X discrete, σ2 = Var[X] = ∑(xi−µ)2 p(xi)

� X continuous, σ2 = Var[X] = ∫ (x−µ)2 f(x)dx

The standard deviationis the (positive) square

root of the variance, σ = (Var[X])1/2 .

Scatter parametersProperty. Var[X] = E[X2]−E[X]2 = E[X2]−µ2

Given a,b∈IR and a random variable X, we have


the following properties of the variance

1. Var[b] = 0 ;

2. Var[aX] = a2Var[X] ;

3. Var[aX+b] = a2Var[X] .

Shape parametersDescribe features from the distribution of a r.v.

different from location and scatter

k-th moment about the origin, m = E[Xk]


k-th moment about the origin, mk = E[Xk]

k-th moment about the mean, µk = E[(X−µ)k]

� Skewness. Skew = µ3/σ3

� Kurtosis. Kurt = µ4/σ4−3

Independent random variablesTwo random variables X and Yare independent

if for any A,B⊂IR,

P((X∈A)∩(Y∈B)) = P(X∈A)P(Y∈B)


P((X∈A)∩(Y∈B)) = P(X∈A)P(Y∈B)

Equivalently, for any x,y∈IR

P((X ≤ x)∩(Y ≤ y)) = P(X ≤ x)P(Y ≤ y)

Property. If X and Yare independent,

Var[X+Y] = Var[X]+Var[Y]

Example(solidmissilefuel)An important factor in solid missile fuel is the particle size.

Significant problems occur if the particle sizes are too large.

From production data in the past, it has been determined that

the particlesize (X micrometers) distribution is the particlesize (X micrometers) distribution is

characterized by


>

=otherwise0

1 if)(

2 xxkxfX

Example(solidmissilefuel)� Compute the value of k so that fX(x) is a valid density

function.

� Find the cumulative distribution function FX(x).

� What size x is exceeded by 50% of the particles?� What size x is exceeded by 50% of the particles?

� What is the probability that the size of a random particle from the manufactured fuel exceeds 4 micrometers?

� What is the probability that the size of a random particle from the manufactured fuel is lower than 5 if we know that it exceeds 4 micrometers?


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Random variables - UC3Mhalweb.uc3m.es/esp/Personal/personas/icascos/esp/2rvariables.pdf · 2. Discrete and continuous random variables Discrete random variables (probability mass