Review some statistical distributions and characteristics

Probability density function

moment generating function,

cumulant generating functions

Probability Theory: Sets

Characteristic function is defined as an expectation value of the function - e(itx)

Moment generating function is defined as (an expectation of e(tx)):

Moments can be calculated in the following way. Obtain derivative of M(t) and take the value of it at t=0

Cumulant generting function is defined as logarithm of the characteristic function

dxxfitxetC )()()(

dxxftxetM )()()(

0

)()(

t

n

nn

dt

tMdxE

))(log(... tCfgc

Review : Assignment ( 1 )

Write all p.d.f , c.d.f and properties for all discrete and continuous distribution you study in STAT 211.

There are many random experiments that involve more than one random variable. For example, an educator may study the joint behavior of grades and time devoted to study; a physician may study the joint behavior of blood

pressure and weight. Similarly an economist may study the joint behavior of business volume and profit. In fact, most real problems we come across will have more than one underlying random variable of interest.

TWO RANDOM VARIABLES

على تعتمد التي العشوائية التجارب من العديد هناكمتغيرعشوائي أكثر

المثال سبيل على ،المخصص -1 والوقت الطالب درجات بين العالقة دراسة

.للدراسةوالوزن -2 الدم ضغط بين العالقة .دراسة

والربح -3 األعمال حجم بين العالقة .دراسةأكثر وجود تشمل الحقيقية المشاكل معظم فإن الواقع، في

عشوائي متغير من

TWO RANDOM VARIABLES

Bivariate Discrete Random VariableDiscrete Bivariate Distribution

التوزيعات الثنائية المنفصلة

Bivariate Discrete Random Variables In this section, we develop all the necessary terminologies for studying bivariate discrete random variables.

Definition 7.1.p 186: A discrete bivariate random variable (X, Y ) is an ordered pair of discrete random variables.

المشترك االحتمالي joint probabilityالتوزيعdistribution المنفصلين دالة )X,Y(للمتغيرين هو

مشتركة قيم f(x,y)احتمالية إحتماالت )X,Y(تعطيجدول صورة في الدالة هذه وتعرض المختلفة

قيم تبين رياضية صيغة في أو )X,Y(مستطيلالدالة هذه وتعرف القيم هذه واحتماالت المختلفة

كاآلتي:

: تحقق الدالة وهذه

TWO RANDOM VARIABLES

(1) ( , ) 0 ( , )

(2) ( , ) 1

XY

XYx y

f x y for all x y

f x y

( , ) ( , )XYf x y P X x Y y

Defintion 7.4. p192

Let (X, Y ) be any two discrete bivariate random variable. The real valued function F is called the joint cumulative probability distribution function of X and Y if and only if

( , ) ( , )

( , )s x t y

F x y P X x Y y

f s t

( , ) ( , )b d

x aY c

P a X b c Y d f x y

Example:

Roll a pair of unbiased dice. If X denotes the sum of Points that appear on the upper surface for the two dice and Y denotes the largest points on the dice. Find the joint distribution for X, Y

كانت فإذا واحدة، مرة نرد زهرتي مجموع xألقيت هيللزهرتين . العلوي السطح على تظهر التي النقاط

للمتغيرين المشتركة االحتمالية الدالة X, Yأوجد

Bivariate Discrete Random Variable

(Joint discrete distribution)

S ={ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

Solution: نرد زهرتي إللقاء العينة :فراغ

Probability Theory

YX

1 2 3 4 5 6

2 1/36 0 0 0 0 0

3 0 2/36 0 0 0 0

4 0 1/36 2/36 0 0 0

5 0 0 2/36 2/36 0 0

6 0 0 1/36 2/36 2/36 0

7

8

Probability Theory

YX

1 2 3 4 5 6

2

3

4

5

6

7

8

Probability Theory

F (4 , 3 ) = P ( X ≤ 4 , Y ≤ 3 )

= 1/36 + 2/36 + 1/36 + 1/36 = 6/36

P ( 6 ≤ X ≤ 8 , 4 ≤ Y < 6 )

= 2/36 + 2/36 + 2/36 + 2/36 + 1/36 + 2/36 = 11/36

Probability Theory

EXAMPLE: If the probability joint distribution for X, Y is given as:

1- Show that f(x,y) is probability mass function?

2- Find f( 2, 4 ).

3- Find F( 2,4).

2 2( , ) ; 1, 2,..., 1

2,3,4,....,

0 1, 1

yf x y q p x y

y

p q p

Solution: :

Solution: :

Definition 7.3. p188 : Let (X, Y ) be a discrete bivariate random variable. Let and be the range spaces of X and Y , respectively. Let f(x, y) be the joint probability density function of X and Y . The function

is called Marginal probability density function of X. Similarly, the function

Marginal probability density function of X :

XR YR

1 ( ) ( , )yy R

f x f x y

Marginal probability density function of Y :

Similarly, the function

is called Marginal probability density function of Y.

2 ( ) ( , )xx R

f y f x y

Probability Theory

Example : Let X and Y be discrete random variables with joint probability density function f(x, y) = ( 1/21 (x + y)

if x = 1, 2;

y = 1, 2, 3

0 otherwise.

What are the marginal probability density functions of X and Y ?

Marginal probability Mass function Examples:

Marginal probability density function Example:

Probability Theory

EXAMPLE: If the probability joint distribution for X, Y is given as:

2- Find f( x).

3- Find f( y).

2 2( , ) ; 1, 2,..., 1

2,3,4,....,

0 1, 1

yf x y q p x y

y

p q p

Marginal probability density function Example:

Marginal probability density function Example:

Theorem: 7.1. p 191

A real valued function f of two variables is joint probability density function of a pair of discrete random variables X and Y if and only if :

(1) ( , ) 0 ( , )

(2) ( , ) 1

XY

XYx y

f x y for all x y

f x y

Example:7.1 page 191

For what value of the constant k the function

given by

Is a joint probability density function of some random variables X , Y ?

1,2,3; 1,2,3( , )

0

k x y if x yf x y

otherwise

Marginal probability density function Example:

Probability Theory