chapter 3- multiple random variables-updated

8/19/2019 Chapter 3- Multiple Random Variables-Updated

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CEN 343 Chapter 3: Multiple random variables

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1. Vector Random Variables

•

Let two random variables X with value x and Y with value y are defined on a sample space S,then the random point (x, y) is a random vector in the XY plane.

• In the general case where N r.v's. X1, X2, . . . XN are defined on a sample space S, they

become N-dimensional random vector or N-dimensional r.v.

2. Joint distribution

•

Let two events = { ≤ } = { ≤ }, ( X and Y two random variables ) withprobability distribution functions () and (), respectively:

() = ( ≤ ) (1) () = ( ≤ ) (2)

• The probability of the joint event { ≤ , ≤ } , which is a function of the members and is called the joint probability distribution function { ≤ , ≤ } = ( ∩ ). It isgiven as follows:

,(,) = ( ≤ , ≤ ) (3)

• If X and Y are two discrete random variables, where X have N possible values and Y haveM possible values , then:

,(,) = ∑ ∑ (,)( )( )=1=1 (4)

• If X and Y are two continues random variables, then:

Probability of the joint

event { = , = } Unit step function


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,(,) = ,(, )

−

− (5)

Example 1-a:

Sol: a-


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b-

Example 1-b:

Let X and Y be two discrete random variables. Let us assume that the joint space has only three

possible elements (1, 1), (2, 1) and (3, 3). The probabilities of these events are:

(1,1) = 0.2, (2, 1) = 0.3, (3, 3) = 0.5. Find and plot ,(, ). Solution:

Proprieties of the joint distribution ,(,):1. ,(∞,∞) = ,(∞,) = ,(,∞) = 2.

,(∞,∞) = 3. ≤ ,(,) ≤ 4. {


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5. ,(,) is a non decreasing function of both and Marginal distributions:

•

The marginal distribution functions of one random variable is expressed as:

,(,∞) = () ,(∞,) = ()

Example 1-c

Suppose that we have two random variables X and Y with joint density function:

Sol:

=13/160

Example 2:

= {(1,1), (2,1), (3,3)}


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(1,1) = 0.2, (2,1) = 0.3, (3,3) = 0.5 Find ,(,) and the marginal distributions X() and Y() of example 1 Solution:

3. Joint density

•

For two continuous random variables X and Y, the joint probability density function is given

by:

,(,) = 2,(,) (6)


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• If X and Y, are tow discrete random variables then the joint probability density function is

given by:

,(,) = (,)( )( )=1

=1

(7)

Proprieties of the joint density function ,(,):1. ,(,) ≥ 0 2.

∫ ∫ ,(,) = 1−− 3. ,(, ) = ∫ ∫ ,(,)−− 4.

() = ∫ ∫ ,(,)−− , () = ∫ ∫ ,(,)−− 5. {1


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In tabular form, the marginal distributions of X and Y are:

Example:

Note that g(x) stands for f X(x) and h(y) for f Y(y). By definition, we have:

Note that, we have:

Example:


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• For discrete random variables:

Suppose we have:

(y) =

=1 (12)

,(,) = ∑ ∑ ,( )=1=1 (13) Assume = is the specific value of y, then:

(| = ) = ( ,)

()

=1 ( ) (14) and (| = ) = (,)()

=1

( ) (15) Example 4:

Let us consider the joint pdf: ,(,) = ()() −(+1)


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Find (|) if the marginal pdf of X is given by: () = ()− Solution:

,(,) and () are nonzero only for > 0 > 0, (|) is nonzero only for > 0 > 0, therefore, we keep u(x).

Calculate the probability: Pr( ≤ | = )? ( ≤ | = ) = ∫ (| = ) =∫ − = − = .

Example 5:

What represents the corresponding figure ?

Find (| = 3)

Solution:

Example:


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Example:

Sol:


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4.2 Interval conditioning: The Radom variable in the interval {


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Show that: ∫ ∫ ,(, )− = ,(, ) ,(, )

Example 6:

Let ,(, ) = ()() −(+1) (18) () = ()−, () = ()(+1)

Find (| ≤ )?Solution 6:

Numerical application: Find (


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• Note that if X and Y are independent, then

(

|

) =

(

)

(

|

) =

(

) (21)

Example:

Example 7:

In example 6, are X and Y independent?

Solution

⟹ . Example 8:

Let ,(, ) = 112 ()()−4− are X and Y independent?

Solution


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Example:

Sol:

6. Sum of two random variables

• Here the problem is to determine the probability density function of the sum of two

independent random variables X and Y :

= + (22) • The resulting probability density function of can be shown to be the convolution of the

density functions of X and Y:

() = () ∗ ()

() = () ( )

−= () ( )

− (23)

Convolution


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Proof:

Example 9:

Let us consider two independent random variables X and Y with the following pdfs:

() = 1 [() ( )]

() = 1 [() ( )] Where 0


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7. Central Limit Theorem

The probability distribution function of the sum of a large number of random variables

approaches a Gaussian distribution. More details will be given in Chapter 5.

a+b

()

w

1

a b


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8. Expectations and Correlations

• If (,) is a function of two random variables X and Y ( their joint probability densityfunction

,(

,

)). Then the expected value of

(

,

), a function of the two random

variables X and Y is given by:

[( ,)] = (,) ,(,)∞

−∞

∞

−∞ ()

Example 10

Let ( ,) = + , find [( ,)]? Solution:

• If we have n functions of random variables (,),(,), … , (,) :

[(,) + (,) + … + (,)] = [(,)] + [(,)] + ⋯+ [(,)]


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Which means that the expected value of the sum of the functions is equal to the sum of the

expected values of the functions.

8.1 Joint Moments about the origin

= [ ] =

−

− ,(,) (25)

We have the following proprieties:

0 = [ ] are the moments of X

0 = [] are the moments of Y The sum n+k is called the order of the moments. ( 20,02 and 11 are called second

order moments).

10=

[

] and

01=

[

] are called first order moments.

8.2 Correlation

• The second-order joint moment 11 is called the correlation between X and Y .• The correlation is a very important statistic and its denoted by :


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= [ ] = 11 =

−

− ,(,) (26)

• If

=

[

]

[

] then X and Y are said to be uncorrelated.

• If X and Y are independent then ,(x, y) = () () and

= ()∞

− ()

∞

−= [ ][] (27)

• Therefore, if X and Y are independent ℎ they are uncorrelated. • However, if X and Y are uncorrelated, it is not necessary that they are independent.

•

If = 0 ℎ Y are called orthogonal.

Example 11

Let [ ] = 3 ,2 = 2, let also = 6 + 22Find ? Are X and Y Orthogonal?

Are X and Y uncorrelated?

Solution:

Or X and Y are correlated.

Example:


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Find the marginal PDF: (), |(|), and E(XY)Solution:

1. () = √ −

2. |(|) = −−

3. ( ) = ∫

√

−− =

8.3 Joint central moments

• The joint central moment is defined as:

= [( �)( �)]= ( �)( �)

−

− ,(,) (28)

We note that:20 = [( �)2] = 2 is the variance of X.2 = [( �)2] = 2 is the variance of Y .

8.4 Covariance

• The second order joint central moment is called the covariance of X and Y and denoted by .

= 11 = [( �)( �)] = ∫ ∫ ( �)( �)−− ,(,) (29) We have

= [ ] �� = [ ][] (30) • If X and Y are uncorrelated (or independent), then = 0 • If X and Y are orthogonal, then = ��


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• If X and Y are correlated, then the correlation coefficient measures the degree of correlation between X and Y:

= =

= [

−�

−� ] (31)

It is important to note that: 1 ≤ ≤ 1 Example 12

Let = + find 2 when X and Y are uncorrelatedSolution:

9 Joint Characteristic functions

• The joint characteristic function of two random variables is defined by:

∅,(1,2)=+ = ∫ ∫ − ,(,)− (32) 1 and 2 are real numbers.•

By setting 1 = 0 2 = 0, we obtain the marginal characteristic function:∅(1) = ∅,(1, 0)

∅(2) = ∅,(0,2) (33)

• The joint moments are obtained as :


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= ()+ ∅,(,) | =0,=0 (34)

Example 1Let us consider the joint characteristic function: ∅ ,(,) = −− Find E [ X ] and E [Y ]

Are X and Y uncorrelated?

Solution:

For = = , the joint characteristic function of: Z=X+Y is:


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chapter 3- multiple random variables-updated

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