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

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

هناك العديد من التجارب العشوائية التي تعتمد على أكثر متغيرعشوائي على سبيل المثال، 1- دراسة العلاقة بين درجات الطالب والوقت المخصص للدراسة. 2- دراسة العلاقة بين ضغط الدم والوزن. 3- دراسة العلاقة بين حجم الأعمال والربح. في الواقع، فإن معظم المشاكل الحقيقية تشمل وجود أكثر من متغير عشوائي TWO RANDOM VARIABLES

Bivariate Discrete Random Variable Discrete 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.

التوزيع الاحتمالي المشترك joint probability distribution للمتغيرين المنفصلين (X,Y) هو دالة احتمالية مشتركة f(x,y) تعطي إحتمالات قيم (X,Y) المختلفة وتعرض هذه الدالة في صورة جدول مستطيل أو في صيغة رياضية تبين قيم (X,Y) المختلفة واحتمالات هذه القيم وتعرف هذه الدالة كالآتي: وهذه الدالة تحقق: TWO RANDOM VARIABLES

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

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 ألقيت زهرتي نرد مرة واحدة، فإذا كانت x هي مجموع النقاط التي تظهر على السطح العلوي للزهرتين. أوجد الدالة الاحتمالية المشتركة للمتغيرين X, Y 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: فراغ العينة لإلقاء زهرتي نرد :

Probability Theory Y X / / /36 2/ /36 2/ /36 2/36 2/

Probability Theory Y X

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).

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 :

Marginal probability density function of Y : Similarly, the function is called Marginal probability density function of 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).

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 :

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 ?

Marginal probability density function Example:

Probability Theory