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The The randomrandom variablevariable

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ContentsContents

1. Definition2. Distribution and density function3. Specific random variables4. Functions of one random variable5. Mean and variance

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The random variableThe random variable

A random variable (RV) represents a process of assigning to

every outcome of an experiment Ω a number ( )= ωx X

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The random variableThe random variable

such that

Thus a RV is a function with domain the set of experimental

outcomes and range a set of numbers

All RV will be written in capital letters.

Ω→X :

∀ ∈ = ω ω ≤ ∈Αrx , A : X( ) x

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Events generated by RVsEvents generated by RVs

For a given random variable, we are able to answer the following question:

What is the probability that the RV X is less than a given number?

That is, what is the probability of all outcomes such that:

What is the probability that the RV X is between the numbers x1 and x2?

That is, what is the probability of all outcomes such that:( )( )ω < ω ≤1 2P : x X x ?

( )( )ω ω ≤P : X x ?

( )≤P X x ?

ω

( )< ≤1 2P x X x ?

ω

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Example:the sum of the outcomes in a two dice roll experiment

( )( )ω ω =P : X x

X=2 (1,1) 1/36 X=3 (1,2);(2,1) 2/36 X=4 (1,3);(2,2);(3,1) 3/36 X=5 (1,4);(2,3);(3,2);(4,1) 4/36 X=6 (1,5);(2,4);(3,3);(4,2);(5,1) 5/36 X=7 (1,6);(2,5);(3,4);(4,3);(5,2);(6,1) 6/36 X=8 (2,6);(3,5);(4,4);(5,3);(6,2) 5/36 X=9 (3,6);(4,5);(5,4);(6,3) 4/36 X=10 (4,6);(5,5);(6,4) 3/36 X=11 (5,6);(6,5) 2/36 X=12 (6,6) 1/36

( )= ωx X ω

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The distribution functionThe distribution function

Given a number x, we can form the event

This event depends on x; hence its probability is a function of x.

This function is called cumulative distribution function.

( ) ω ω ≤: X x

( ) ( ) ( )( )= ≤ = ω ω ≤ ∀ ∈XF x P X x P : X x x

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Properties of distributionsProperties of distributions

1.The function , is monotonically increasing; that is,

if then

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3.

4.

<1 2x x

( )XF x

( ) ( )≤X 1 X 2F x F x

( ) → → +∞XF x 1 x ( ) → → −∞XF x 0 x

( ) ( ) ( )< ≤ = − ∀ ∈ <1 2 X 2 X 1 1 2 1 2P x X x F x F x x ,x , x x

( ) ( )> = − XP X x 1 F x

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Properties of distributionsProperties of distributions

5. The function might be continuous or discontinuous .

If the experiment consists of finitely many outcomes,

is a staircase function.

This is also true if consists of infinitely many

outcomes but X takes finitely many values.

Ω

Ω

( )XF x

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Properties of distributionsProperties of distributions

Let’s consider a discontinuous function, for example that of the sum of two dice roll:

0

0,2

0,4

0,6

0,8

1

1,21 2 3 4 5 6 7 8 9 10 11 12

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Properties of distributionsProperties of distributions

Let’s examine its behavior at a discontinuous point, e.g. at the value x0 = 3.We have:

0

0,2

0,4

0,6

0,8

1

1,2

1 2 3 4 5 6 7 8 9 10 11 12

( ) ( )−→

= < =Xx 3

1lim F x P X 336

( ) ( )→

= ≤ =Xx 3

3limF x P X 336

( ) ( )+→

= ≤ =Xx 3

3lim F x P X 336

( ) ( ) ( ) ( )+ −→ →

− = ≤ − < = = =X Xx 3 x 3

2lim F x lim F x P X 3 P X 3 P(x 3)36

This is true in general.

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Continuous and Discrete RVsContinuous and Discrete RVs

We shall say that a RV X is of continuous type if its

distribution is continuous for every x.

In this case

Hence,

( ) ( ) ( )− +→ →

= =0 0

X X 0 Xx x x xlim F x F x lim F x

( )= =0P X x 0

( ) ( ) ( )≤ = < = XP X x P X x F x

( ) ( ) ( ) ( )< ≤ = < < = −1 2 1 2 X 2 X 1P x X x P x X x F x F x

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Continuous and Discrete RVsContinuous and Discrete RVs

We shall say that a RV X is of discrete type if its

distribution is a staircase function.

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The density function of continuous RVsThe density function of continuous RVs

We can use the distribution function to determine the

probability that the RV X takes values in an arbitrary region R

of the real axis.

This result can be expressed in terms of the derivative

f(x) of F(x), assuming that the derivative of F(x) exists nearly

everywhere.

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The density function of continuous RVsThe density function of continuous RVs

The derivative of the distribution function is called

probability density function

( ) ( ) ( ) ( ) ( )∆ → ∆ →

+ ∆ − < ≤ + ∆′= = =

∆ ∆X X

X X x 0 x 0

F x x F x P x X x xf x F x lim lim

x x

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Properties of density functionProperties of density function

1. Since F increases as x increases, we conclude that

2. Integrating the density function from x1 to x2 we obtain

Thus the area of fX(x) in an interval (x1,x2) equals the

probability that X is in this interval.

( ) ≥Xf x 0

( ) ( ) ( ) ( )= − = < <∫2

1

x

X X 2 X 1 1 2x

f x dx F x F x P x X x

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Properties of density functionProperties of density function

3.

4. Finally, which is referred to as the

normalization condition.

( ) ( )∞

= ∫x

X XF x f x dx

( )+∞

−∞

=∫ Xf x dx 1

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The probability function of a discrete RVsThe probability function of a discrete RVs

Suppose now that F is a staircase function with discontinuities at the points xi.

In this case, the RV takes the values xi with probability:

The numbers pi will be represented graphically by vertical segments at the points xi with height equal to pi.

( ) ( ) ( )−= = = −i i X i X iP X x p F x F x

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The probability function of a discrete RVsThe probability function of a discrete RVs

Occasionally, we shall use the notation

The function fX so defined will be called point density.

Note that its values are not the derivatives of FX, but they equal the discontinuity jumps of FX.

( )=i X ip f x

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SpecificSpecific randomrandom variablesvariables

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We introduce various RVs with specified distribution.

To do so it is not necessary to specify the underlying experiment.

Given a distribution F, we can construct an experiment and a RV X such that its distribution equals F.

In the study of a RV we can avoid the notion of an abstract space.

We can assume that the underlining experiment is the real line and its outcomes the value x of X.

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Discrete Discrete randomrandom variablesvariables

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The The bernoullibernoulli RVRV

We shall say that a RV X has a Bernoulli distributionif it assumes the following two values with the corresponding probability

⎧= ⎨ −⎩

0 1X

1 p p

0 1 x

fX

1-pp

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The Bernoulli RVThe Bernoulli RV

The corresponding distribution is

( )<⎧

⎪= − ≤ <⎨⎪ ≥⎩

X

0 X 0F x 1 p 0 X 1

1 X 1

FX

11-p

0 1 x

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The binomial RVThe binomial RV

We shall say that a RV X has a Binomial distribution of order n if takes the values 0,1,…n with probabilities

( ) −⎛ ⎞= = = + =⎜ ⎟

⎝ ⎠k n kn

P X k p q k 0,1,...,n; p q 1k

The Binomial distribution originates in the experiment of independently repeated trials, if we define X as the number ofsuccesses of an event A (P(A)=p) in n trials.

Assume we toss n times a fair coin. A is the event ‘head’, whose probability is p=0.5.X is the number of heads in the n repetitions .

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Continuous random variablesContinuous random variables

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The Uniform RVThe Uniform RV

We shall say that a RV X is uniform or uniformly distributed in the interval (a,b) if

( )⎧ ≤ ≤⎪= −⎨⎪⎩

X

1 a X bf x b a

0 elsewhere

a b

1b a−

x

fX

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The Uniform RVThe Uniform RV

The corresponding distribution is a ramp

( )

<⎧⎪⎪ −= ≤ ≤⎨

−⎪⎪ >⎩

X

0 X ax aF x a X bb a1 X b

a b

1

x

FX

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The Exponential RVThe Exponential RV

We shall say that a RV X has an exponential distribution if

( ) −

<⎧= ⎨

> >⎩X cx

0 x 0f x

ce x 0,c 0

c

x

fX

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The Uniform RVThe Uniform RV

The corresponding distribution is

( ) −

<⎧= ⎨

− ≥ >⎩X cx

0 x 0F x

1 e x 0,c 0

1

x

FX

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The Normal RVThe Normal RV

We shall say that a RV X is standard normal or Gaussian if its density is the function

( )−

=π

2x2

X1g x e2

x

1

gX

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The Normal RVThe Normal RV

The corresponding distribution is the function

( )ξ

−

−∞

= ξπ∫

2x2

X1G x e d2

x

1GX

From the evenness of g(x) it follows that ( ) ( )− = −X XG x 1 G x

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The Normal RVThe Normal RV

Shifting and scaling g(x), we obtain the general normal curves:

( )( )−µ

−σ − µ⎛ ⎞= = ⎜ ⎟σ σσ π ⎝ ⎠

2

2x2

X1 1 xf x e g2

σ

x

fX

µ

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The Normal RVThe Normal RV

We shall use the notation to indicate that the RV X is normal.

Thus indicates that X is standard normal.

If X is normal, , then [ ]µ σ∼X N ,

[ ]∼X N 0,1

[ ]µ σ∼X N ,

( ) ( ) ( ) − µ − µ⎛ ⎞ ⎛ ⎞< < = − = −⎜ ⎟ ⎜ ⎟σ σ⎝ ⎠ ⎝ ⎠2 1

1 2 X 2 X 1 X Xx xP x X x F x F x G G

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The Normal RVThe Normal RV

With

one has

This is the area of the normal curve in the interval .

The following special cases are of particular intrests:

( )µ − σ µ + σk , k

( ) ( ) ( ) ( )µ − σ < < µ + σ = − − = −X X XP k X k G k G k 2G k 1

= µ − σ = µ + σ1 2x k , x k

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The Normal RVThe Normal RV

The following special cases are of particular interests:

( )µ − σ < < µ + σP X 0.683

( )µ − σ < < µ + σP 2 X 2 0.954

( )µ − σ < < µ + σP 3 X 3 0.997

We note further that:

( )µ − σ < < µ + σ =P 1.96 X 1.96 0.95

( )µ − σ < < µ + σ =P 2.58 X 2.58 0.99

( )µ − σ < < µ + σ =P 3.29 X 3.29 0.999

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Functions of one random variableFunctions of one random variable

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Functions of one random variableFunctions of one random variable

Given a function g(x), of the real variable x and a RV X with domain the space Ω, we form the composite function

This function define a RV Y with domain the set Ω .

For a specific the value of the RV so formed is

given by , where .

( )ωiYω ∈Ωi

( ) ( )( )ω = ωY g X

( )=i iy g x

x

( )= ωi ix X

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Density of Density of g(Xg(X))

It is possible to determine the density fY of the RV Y=g(X) in terms of the density fY of the RV X.

To do so we form the equation

Where y is a specified number, and we solve for x.

( ) =g x y

x

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Density of Density of g(Xg(X))

The solutions of this equation are the abscissas xi of the intersections points of the horizontal line LY with the curve g(x)

y Ly

x1 x2 x3

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Density of Density of g(Xg(X))

Theorem:For a specific y, the density fY(y) is given by:

where xi are the roots of g(x)=y, and g’(xi) are the derivatives of g(x) at x=xi.

( ) ( )( )=

=′∑ i

Y ii

f x xf y

g x