Fundamentals of Probability and Statistics for Engineers

3.13 For each of the joint probability mass functions (jpmf), pX Y (x,y), or joint prob-
ability density functions (j pdf), fX Y (x, y), given below (cases 1–4), determine:
(a) the marginal mass or density functions,
(b) whether the random variables are independent.

(i) Case 1

(ii) Case 2:

(iii) Case 3

(iv) Case 4

3.14 Suppose X and Y have jp mf

(a) Determine marginal pmfs of X and Y.

(b) Determine P(X 1).

(c) Determine P(2X Y).

3.15 Let X 1 ,X 2 ,andX 3 be independent random variables, each taking values 1 with
probabilities 1/2. Define ra ndom variables Y 1 ,Y 2 ,andY 3 by

Show that any two of these new random variables are independent but that Y 1 ,Y 2 ,

and Y 3 are not independent.

3.16 The random variables X and Y are distributed according to the jpdf given by
Case 2, in Problem 3.13(ii). D etermine:
(a)
(b)

Random Variables and Probability D istributions 71

pXY…x;y†ˆ

0 : 5 ; for…x;y†ˆ… 1 ; 1 †;

0 : 1 ; for…x;y†ˆ… 1 ; 2 †;

0 : 1 ; for…x;y†ˆ… 2 ; 1 †;

0 : 3 ; for…x;y†ˆ… 2 ; 2 †:

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fXY…x;y†ˆ

a…x‡y†; for 0<x 1 ;and 1<y 2 ;

0 ; elsewhere:



fXY…x;y†ˆ

e…x‡y†; for…x;y†>… 0 ; 0 †;

0 ; elsewhere:



fXY…x;y†ˆ

4 y…xy†e…x‡y†; for 0<x< 1 ;and 0<yx;

0 ; elsewhere:



pXY…x;y†ˆ

0 : 1 ; for…x;y†ˆ… 1 ; 1 †;

0 : 2 ; for…x;y†ˆ… 1 ; 2 †;

0 : 3 ; for…x;y†ˆ… 2 ; 1 †;

0 : 4 ; for…x;y†ˆ… 2 ; 2 †:

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Y 1 ˆX 1 X 2 ; Y 2 ˆX 1 X 3 ; Y 3 ˆX 2 X 3

P 9 X 0 : 5 \Y> 1 :0).

P 9 XY<^12 ).