Fundamentals of Probability and Statistics for Engineers

Problems

5.1 Determine the Probability distribution function (PDF) of Y 3 X 1 if
(a) Case 1:

(b) Case 2:

5.2 Temperature C measured in degrees Celsius is related to temperature X in degrees
Fahrenheit by C 5(X 32)/9. Determine the probability density function (pdf) of
C if X is random and is distributed uniformly in the interval (86, 95).

5.3 The random variable X has a triangular distribution as shown in Figure 5.22.
Determine the pdf of Y 3 X 2.

5.4 Determine FY (y) in terms of FX (x) if Y X1/2, where FX (x) 0, x < 0.

5.5 A random variable Y has a ‘log-normal’ distribution if it is related to X by Y eX,
where X is normally distributed according to

Determine the pdf of Y for m 0 and 1.

1

–1 1

x

fX(x)

Figure 5. 22 Distribution of X, for Problem 5.3

154 Fundamentals of Probability and Statistics for Engineers

ˆ

FX…x†ˆ

0 ; forx< 3 ;

1

3

; for 3x< 6 ;

1 ; forx 6 :

8

>>

<

>>:

FX…x†ˆ

0 ; forx< 3 ;

x

3



1 ; for 3x< 6 ;

1 ; forx 6 :

8

><

>:

ˆ

ˆ‡

ˆ ˆ

ˆ

fX…x†ˆ

1

… 2 †^1 =^2

exp

…x m†^2

2 ^2

"#

; 

1 <x< 1

ˆ ˆ