Page 88 - Fundamentals of Probability and Statistics for Engineers

Random Variables and Probability Distributions 71
3.13 For each of the joint probability mass functions (jpmf), p XY (x, y), or joint prob-
ability density functions (jpdf), f XY (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

0:5; for x; y†ˆ 1; 1†;
8
>
>
0:1; for x; y†ˆ 1; 2†;
>
<
p XY x; y†ˆ
0:1; for x; y†ˆ 2; 1†;
>
>
>
:
0:3; for x; y†ˆ 2; 2†:
(ii) Case 2:
a x ‡ y†; for 0 < x 1; and 1 < y 2;

f x; y†ˆ
XY
0; elsewhere:
(iii) Case 3
x‡y†
e ; for x; y† > 0; 0†;
f XY x; y†ˆ
0; elsewhere:
(iv) Case 4
x‡y†
4y x y†e ; for 0 < x < 1; and 0 < y x;
f XY x; y†ˆ
0; elsewhere:
3.14 Suppose X and Y have jpmf
0:1; for x; y†ˆ 1; 1†;
8
>
>
>
< 0:2; for x; y†ˆ 1; 2†;
p XY x; y†ˆ
0:3; for x; y†ˆ 2; 1†;
>
>
>
:
0:4; for x; y†ˆ 2; 2†:
(a) Determine marginal pmfs of X and Y.
(b) Determine P(X ˆ 1).
(c) Determine P(2X Y ).
3.15 Let X 1 , X 2 , and X 3 be independent random variables, each taking values 1 with
probabilities 1/2. Define random variables Y 1 , Y 2 , and Y 3 by
Y 1 ˆ X 1 X 2 ; Y 2 ˆ X 1 X 3 ; Y 3 ˆ X 2 X 3
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). Determine:
(a) P9X 0:5 \ Y > 1:0).
1
(b) P9XY < ).
2

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