Page 171 - Fundamentals of Probability and Statistics for Engineers

154 Fundamentals of Probability and Statistics for Engineers

PROBLEMS
5.1 Determine the Probability distribution function (PDF) of Y ˆ 3X
1 if
(a) Case 1:
0; for x < 3;
8
>
>
1
<
F X x†ˆ ; for 3 x < 6;
3
>
>
1; for x 6:
:
(b) Case 2:
0; for x < 3;
8
>
< x
F X x†ˆ
1; for 3 x < 6;
3
>
:
1; for x 6:
5.2 Temperature C measured in degrees Celsius is related to temperature X in degrees
Fahrenheit by C ˆ 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 ˆ 3X ‡ 2.
5.4 Determine F Y (y) in terms of F X (x) if Y ˆ X 1/2 , where F X (x) ˆ 0, x < 0.
X
5.5 A random variable Y has a ‘log-normal’ distribution if it is related to X by Y ˆ e ,
where X is normally distributed according to
" #
1
x
m† 2
f x†ˆ 1=2 exp 2 ;
1 < x < 1
X
2 † 2
Determine the pdf of Y for m ˆ 0 and ˆ 1.
(x)
f X

1

x
–1 1

Figure 5.22 Distribution of X, for Problem 5.3

TLFeBOOK