Page 84 - Fundamentals of Probability and Statistics for Engineers

Random Variables and Probability Distributions 67

PROBLEMS
3.1 For each of the functions given below, determine constant a so that it possesses all
the properties of a probability distribution function (PDF). Determine, in each case,
its associated probability density function (pdf) or probability mass function (pmf)
if it exists and sketch all functions.
(a) Case 1:

0; for x < 5;
F x†ˆ
a; for x 5:
(b) Case 2:
8
0; for x < 5;
>
1
<
F x†ˆ ; for 5 x < 7;
3
>
a; for x 7:
:
(c) Case 3:
8
0; for x < 1;
>
k
<
F x†ˆ X j
1=a ; for k x < k ‡ 1; and k ˆ 1; 2; 3; ... :
>
:
jˆ1
(d) Case 4:

0; for x 0;
F x†ˆ
1 e ax ; for x > 0:
(e) Case 5:
8
0; for x < 0;
<
a
F x†ˆ x ; for 0 x 1;
1; for x > 1:
:
(f) Case 6:
8
0; for x < 0;
< p

F x†ˆ a sin 1 x; for 0 x 1;
1; for x > 0:
:
(g) Case 7:
0; for x < 0;

F x†ˆ 1
a 1 e x=2 †‡ ; for x 0:
2
3.2 For each part of Problem 3.1, determine:
(a) P(X 6);
1
(b) P( < X 7).
2

TLFeBOOK