MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI

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1 MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI QUESTION BANK - ANSWERS SEMESTER: IV MA - PROBABILITY AND QUEUEING THEORY UNIT II: TWO DIMENSIONAL RANDOM VARIABLES PART-A Question : AUC M / J If the joint pdf of ( X,) Y is given b ( f,),in then find () E X f () ( f,) d d () E() X () f d Question : AUC M / J,M / J, State central limit theorem Let X, X,...be a sequence of independent and identicall distributed random variables each having mean and variance.then the distr tends to the standard normal as Question : AUC M / J n ibution of X X... X n n n The regression equations of X on Y and Y on X are respectivel 5 and 5.Find the means of X () () 5 and Put in () we get Hence X Y Y

2 Question : AUC M / J Given the two regression lines X Y 9 and X 9Y, find the coefficient of correlation between X and Y 9 9 b 9 b 9 r bb r.9 Question : AUC N / D When will the two regression lines be (a). a right angles (b). coincident (). a If r (). If b r Question : 5 AUC N / D 5 A small college has 9 male and female professors. An ad-hoc committee of 5 selected at random to unite the vision and mission of the college. If X and Y are the number of men and women in the committee, respectivel,what is the joint probabilit mass funcion of X and Y 9C C P,, 9, C Question : AUC N / D The joint pdf of the RV (X,Y) Ke dd is ( f,) Ke,,.Find K Put K e d e d t dt d d t dt t dt K e e K t t e dt e dt K dt

3 Question : 7 AUC N / D Given the RV with densit function () f,.fin the pdf of, otherwise Y X Answer d f ()() f, d

4 Question : AUC M / J PART B The joint pdf of the two dimensional RV is ( f,),, Find. (). P X Y ( ). P X Y (). P/ X Y (). P X Y dd d 9 d 9 d 9 7 d d (). P X Y () d dd d d d 9 d d (9) d 7 9 d 7 d P X Y (). P X / Y P( Y ) f ()(, f ) d d 5

5 5 P( Y )() f d5 d P X Y (). P X / Y P( Y ) 5 5 Question : AUC M / J i j Let X and Y be two random variables having the joint pdf ( f,)( ), k wh ere and can assume onl the integer values, and. Find the marginal and conditional distributions. f (,)( k) f (,) i j k k k k 5k k k k k k k k 5k k k k k 7 P() P( X ) P( X ) P( X ) P() 9 5 P( Y ) P( Y ) P( Y ) Conditional distribution of X, given Y PY P X, Y P P X / Y P Y 7 P X, Y P P X / Y 7 PY PY 7 5

6 P X, Y P P X / Y 7 PY PY 7 P X, Y P P X / Y 7 PY PY 7 P X, Y P P X / Y 7 PY PY 7 P X, Y P P X / Y 7 PY PY 7 P X, Y P P X / Y 7 PY PY P X, Y P P X / Y 7 PY PY P X, Y P P X / Y 7 PY PY Conditional distribution of Y, given X P X P X, Y P PY / X P X 7 P X, Y P PY / X 7 P X P X 7 PY, X P PY / X 7 P X P X 7 PY, X P PY / X 7 P X P X 7 P X, Y P PY / X 7 P X P X 7

7 5 PY, X P 5 PY / X 7 P X P X 7 PY, X P PY / X 7 P X P X 7 PY, X P PY / X 7 P X P X 7 P X, Y P P X / Y 7 P X P X 7 Question : AUC M / J Two random variables X and Y have joint probabilit densit function f (,) c(),,, otherwise Find cov X, Y Given (,) f dd c() dd c d c d c d c c (),, Now f(,), otherwise Cov X,()() Y E XY E X E Y f ()() d f ()() d 7

8 5 E()() X f d d d 5 E()() Y f d d d E XY f (,)() dd dd Cov X,()() Y E XY E X E Y Question : AUC M / J Two dimensional random variable X, Y have the JPDF ( f,) (). Find P X Y ().Find the marginal and conditional distributions P X Y dd d 5 f ()(,) f d () d f ()(,) f d d f / f / d f (,) f () f (,) f () Question : 5 AUC M / J,, otherwise Suppose that in a certain circuit, resistors are connected in series. the mean and variance of each resistors are 5 and. respectivel. Using central limittheorem, find the probabilit that the total resistance of the Let X i represents the resistance in the ith resistor. circuit will eceed 9 ohms assuming independence. Then X, X,... X are identicall distributed random variables Let S X X... X n Given E X 5 i

9 V X..5, n X Z n 9 Sn 9 Z.995. Z n P S 9 P Z.5 P Z.5.. Determine whether the random variables X and Y are independent, given their Question : AUC M / J,, joint probabilit densit function as ( f,), otherwise ()(,) f f d d f ()(,) f d d f ()() f Question : 7 AUC M / J f (,) ( ),, Given the joint densit function ( f, ) Find the marginal densities, elsewhere g(),() hand the conditional densit ( / f ) and evaluate P / Y ( ) ()(,)( ) g f d d d ( )( ) ()(,)( ) h f d d d f / ( ) f (,) h() ( ) P / Y d 9

10 Question : AUC M / J If X and Y are independent random variables having densit function e, e, f () and () f, respectivel,,, find the densit function of Z X Y f (,)()() f f e e e XY X Y Given U X Y. Let W Y The transformation functions are u, w Solving we get u w, w (,) u w J ( u,) w u w f ( u,)( w,) J f e f ()( u,) u5w u, if u 5 f u w dw u e u 5, if