Fundamentals of Probability and Statistics for Engineers

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 (PD F ). D etermine, in each case,
its associated probability density function (pdf ) orprobability mass function (pmf)
if it exists and sketch all functions.
(a) Case 1:

(b) Case 2:

(c) Case 3:

(d) Case 4:

(e) Case 5:

(f) Case 6:

(g) Case 7:

3.2 For each part of Problem 3.1, determine:
(a) P(X 6);
(b) P(^12 < X 7).

Random Variables and Probability D istributions 67

F…x†ˆ^0 ; forx<^5 ;

a; forx 5 :



F…x†ˆ

0 ; forx< 5 ;

1

3

; for 5x< 7 ;

a; forx 7 :

8

><

>:

F…x†ˆ

0 ; forx< 1 ;

Xk

jˆ 1

1 =aj; forkx<k‡ 1 ;andkˆ 1 ; 2 ; 3 ;...:

8

><

>:

F…x†ˆ^01 ;eforax;xfor^0 ;x> 0 :



F…x†ˆ

0 ; forx< 0 ;

xa; for 0x 1 ;

1 ; forx> 1 :

8

<

:

F…x†ˆ

0 ; forx< 0 ;

asin^1



x

p

; for 0x 1 ;

1 ; forx> 0 :

8

<

:

F…x†ˆ

0 ; forx< 0 ;

a… 1 ex=^2 †‡

1

2

; forx 0 :