Module 5: Multiple Random Variables. Lecture 1: Joint Probability Distribution

Transcription

1 Module 5: Multile Random Variables Lecture 1: Joint Probabilit Distribution 1. Introduction irst lecture o this module resents the joint distribution unctions o multile (both discrete and continuous) random variables. Joint d and CD o bivariate distributions are articularl discussed with the hel o numerical examles.. Multile Random Variables Previousl the theoretical concets o single random variable have been discussed. In man cases it ma be necessar to deal with more than one random variable within the same exeriment and the same samle sace. In this lecture the theor rom single random variable would be extended to two random variables and then to multile random variables. It ma be recalled that random variable is a unction that mas each oints over a samle sace to a numerical value on the real line. Let us consider two (or more) random variables both (or all) o them maing rom the same samle sace. Accordingl Multile Random Variable ma be deined as ollows. An n -dimensional random vector (i.e. vector o random variables) is a unction that mas n each and ever outcome rom the samle sace S to R ( N dimensional Euclidean sace). (or an non-negative integer n the sace o all n -tules o real numbers orms an n- n dimensional vector sace called N dimensional Euclidean sace over R and denoted b R where R denotes the ield o real numbers). Grahical reresentation o bivariate random variables requires a three-dimensional orm in which the two horizontal axes reresent the two random variables and the m or d is measured verticall. Samle Sace ig. 1. Grahical reresentation o bivariate random variables

2 or simlicit irst two random variables (Bivariate Random Variables) will be considered here i.e n so that our the random vector becomes a ordered air. Bivariate random variables ma be discrete or continuous. 3. Bivariate Random Variables Man real lie situations in the ield o civil engineering require consideration o two or more random variables. or examle average rainall over a catchment area and volume o streamlow assing through the outlet o the catchment over a eriod o time. I other variables such as deth o ground water table is also considered then it becomes multivariate. 4. Probabilit Distribution unction The robabilit o two events A x B Y deined as unctions o x and resectivel are called Cumulative Distribution unctions (CD). x P x PY Y To consider the joint event used. 5. Joint Probabilit Distributions o Discrete Bivariate RV x a concet called joint distribution unction is I and Y are two random variables the joint robabilit distribution o and Y is a descrition o the set o oints x in the range o along with the robabilit o each Y x is given b oint. Joint cumulative distribution unction o and Y denoted b x P x Y The air is reerred as the Bivariate random variable. Joint robabilit distribution is also reerred to as Bivariate Probabilit Distribution or Bivariate Distribution or the case o two random variables and generalized to an number o random variables as Multivariate Distribution. 5.1 Proerties o Joint Distribution unction As a robabilit unction x Y holds certain roerties: x 1 or x 1. Y. x Y is nonnegative and a nondecreasing unction o x and 3. Y ( ) 1; ( ) 4. ( ) ; ( ) Y

3 5. ( x ) ; ( x ) 5. Joint PM and CD o Discrete RV x The joint robabilit mass unction o two discrete random variables describes how much robabilit mass is concentrated on each ossible airs o x. It is given b the intersection robabilit. x P x Y x S The joint cumulative distribution unction is the sum o robabilities associated with all oint airs x i i in the subset x i x i. It is given b 5.3 Proerties o Joint PM x 1 1. Y x xi j xi x; j. x S x 1 3. all x all Y x x Y 5.4 Problem on Joint PM Q. The joint m o two random variables and Y is given b x k(x 5) x 1; 1 otherwise What is value o k? Soln. rom the roerties o joint m x S x 1 x k all x all x1 1 1 Thus k. 4 k(x 5) k 1

4 Q. Streamlows at two gauging stations on two nearb tributaries are categorized into our dierent states i.e. 1 3 and 4. These catagories are reresented b two random variables and Y resectivel or two tributaries. m o streamlow categories and Y are shown in the table on the next slide. Calculate the robabilit o Y. Y 1 Y Y 3 4 Y x Y Soln. Let the robabilit P A reresent the event Y 4 and This will include the set 1 Thus robabilities o these sets should be added u to the required robabilit. Thus according to joint robabilit mass unction the robabilit is given b: x P Y x all ossible x Joint Pd and CD o Continuous Random Variables Let be the continuous random variable then robabilit or the event x1 x and Y is deined b the integration o joint d over the region o interest in the samle 1 sace. P x x x Y x 1 1 x1 1 ddx

5 Grahicall the equation reresents the volume under the joint d x o interest (reer ig ). Y over the region 6.1 Proerties o Joint d x 1. Y. x dxd 1 Y 6. Joint CD o continuous variables ig.. Joint d o continuous RVs The joint distribution o can be comletel described with their joint CD x P x Y x dd ig. 3. Joint CD o continuous RVs

6 The relationshi between jont d and joint CD is given b: x x x It ma be noted that artial derivatives are used in lace o the derivatives or bivariate / multivariate cases. 6.3 Problem on Joint d Q. A storm event occurring at a oint in sace is characterized b two variables namel the duration o the storm and its intensity which is deined as the average rainall rate. The variables and Y are taken to be distributed as ollows: Y x 1 e x x 1 e The joint CD o and Y is assumed to ollow exonential bivariate distribution given b: x x cx x 1 e e e x with c denoting a arameter describing the joint variabilit o the two variates. ind the ossible values that c can take. Soln. The lower and uer boundar o c has to be determined. irst let us determine the lower bound o c. Now P x Y P x x x cx Thus 1 e e e 1 e which gives x 1 c x Since x and are alwas nonnegative the inequalit holds i and onl i 1 c Now let us determine the uer bound o c. x We know x x Dierentiating the CD w.r.t x we have

7 x x x cx 1 e e e x x cx x Now dierentiating the above equation w.r.t or x e x x e 1 c e x cx 1 c cx ce 1 c e x the joint d at the origin is c Y. x cx Since the d is a nonnegative unction the inequalit c must hold; hence the uer bound o arameter c is c. 7. Concluding Remarks Joint ds and CDs o bivariate random variables are discussed in this lecture. Examle roblems on joint distributions are also resented here. The next lecture resents marginal robabilit distributions o discrete and continuous random variables.